diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzpcju" "b/data_all_eng_slimpj/shuffled/split2/finalzzpcju"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzpcju"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n  \nIC 348 is a young (less than 10Myr) and nearby cluster ( distance 316 pcs)\nlocated in the Perseus complex (Lada $\\&$ Lada 1995, Trullols $\\&$\n Jordi 1997, Herbig 1998, Luhman et al. 1998).\nThis cluster  has a number of T Tauri stars\n(Herbig 1954) which are lower mass (lower than 1.5 \\hbox{M$_{\\odot}$}) Pre-Main-Sequence\nstars. Deep near infrared imaging survey of IC 348 in the J, H and K bands by\nLada $\\&$ Lada (1995) resulted with 380 NIR sources as probable cluster \nmembers. Herbig (1998) made a survey for stars having H$_\\alpha$ emission\nand discovered over 110 emission line stars brighter than R= 19 magnitude. \nHe found the proportion of WTTSs (weak line TTSs with H$_\\alpha$ \nequivalent width  below 10\\AA~  and H$_\\alpha$ emission can be assumed\n to be chromospheric origin) to CTTSs (classical TTSs with H$_\\alpha$\nequivalent width above 10\\AA~ and H$_\\alpha$ emission is probably dominated by\nthe accretion of circumstellar material on to the stars)\n as 58:51. CTTSs exhibit infrared excess and \nshow a varying photometric light curves irregularly.\n WTTSs show spectroscopic and photometric periodic variability\non time scales of days caused by rotational modulation due to magnetic\n activity. \nLuhman et al. (1998) performed deep infrared and optical spectroscopy of IC 348\n and found that nearly 25$\\%$ of stars within the core of IC 348 and younger\n than 3 Myr exhibits signature of disks in the form of strong H$_\\alpha$.\n\nHerbst et al. (2000) studied the photometry of 150 stars and discovered \n19 periodic variables with periods ranging from 2.24 to\n16.2 days and masses ranging from 0.35 to 1.1 \\hbox{M$_{\\odot}$}.\nThis variability is caused by the rotation of the surface with large cool spots whose pattern is often stable for many rotation periods.\nRecently Cohen et al. (2004) presented results based on 5 yr of monitoring this cluster and found that these periodic stars show modulations of\n their amplitude, mean brightness and light curve shape on time scales of less than one year.\n\nX-ray observations of IC 348 with ROSAT by Preibisch et al. (1996) resulted with detection of 116 X-ray sources. They found probable new cluster members.\nThey suggested that these were presumably weak line T Tauri stars\nbecause of their X-ray properties. WTTSs seem to be stronger X-ray emitters\nthan the CTTSs. They could not find any significant correlation\nbetween the H$_\\alpha$ luminosity and X-ray luminosity indicating that \nH$_\\alpha$ emission is not a chromospheric emission for CTTSs.\nPreibisch $\\&$ Zinnecker (2001) detected 215 X-ray sources with the Advanced\nCCD Imaging Spectrometer on board the Chandra X-Ray Observatory.\n58 of these sources were identified as new cluster members.\nThey did not find significant differences between the X-ray properties of WTTSs and CTTSs. About 80$\\%$ of cluster members with masses between 0.15 and 2 \\hbox{M$_{\\odot}$}\nwere identified as visible X-ray sources. The observed X-ray emission\n was explained as coronal emission for WTTSs. Chandra X-ray detection\nfraction of the IC 348 cluster was high for spectral types between\nthe late F and M4. In their next study, Preibisch $\\&$ Zinnecker (2002)\nfound a tight correlation between X-ray luminosity and H$_\\alpha$\nluminosity for the WTTSs. They suggested that the chromosphere was heated by\n X-rays from the overlying corona. The CTTSs did not show such a \nrelation since H$_\\alpha$ emission comes mainly from accretion  processes.\nThey pointed out that the use of H$_\\alpha$ emission as an indicator\nfor circumstellar material had some problems. \n\nThe main goal of this study is to find out if there is a variability in\n the light curve of some cluster members which have X-ray counterparts.\nWe wanted to examine correlation between X-ray\nluminosity and rotational period  of stars in this cluster\nin order to see whether rotation is an important \nparameter governing the X-ray emission. We chose\n some X-ray emission sources which were detected and located by Chandra X-Ray\nObservatory. We investigated the corresponding optical \n light curves of these sources obtained by robotic ROTSEIIId \ntelescope in order to search for variability.\nWe also wanted to  investigate the observational results for {$\\delta$ Scuti }  \nstar H254 which is a member of this cluster. \nThis star was previously detected by Ripepi et al. (2002)\nand found as a {$\\delta$ Scuti } star.\n On the basis of observations of Luhman et al. (1998) H254 (L= 31.4\n L$_{\\odot}$ and T$_{e}$= 7200 K) is \n inside the theoretical pulsational\ninstability strip for the PMS stars determined by Marconi and Palla (1998).\nRipepi et al. obtained that H254 pulsates with a pulsation frequency of\n7.406 $d^{-1}$. This frequency was confirmed in our observations.\nRipepi et al.  calculated that this star pulsates either in the fundamental\nmode or in the first overtone. They gave a mass range of 2.3 and 2.6 \\hbox{M$_{\\odot}$}\nfor this star by computing a sequence of linear non-adiabatic models.\n In the second section the observations and\ndata reduction were discussed. The results  and discussion \nrelated with the periodic\n variables in this cluster were given in the \nsection 3. We summarized\nour results in the last section.  \n\n\n\\section{Observations and Data reduction}\n\nThe CCD observations of cluster stars were performed during August, 2004 and \nJanuary, 2005 with ROTSEIIId robotic reflecting telescope \nlocated at the Turkish National Observatory (TUG) site, \nBak{\\i}rl{\\i}tepe, Turkey. ROTSEIII telescopes were described\n in detail by Akerlof et al. (2003). They were designed\nfor fast ($\\sim$6 s) responses to Gamma-Ray Burst triggers from\nsatellites such as Swift.\n It has a 45 cm mirror and operates without filters.\nIt has equipped with a CCD, 2048$\\times$2048 pixel,\n the pixel scale is 3.3 arcsec\nper pixel for a total field of view\n 1.$^{\\circ}$85$\\times$1.$^{\\circ}$85. A total of about 1800 CCD\n frames were collected during the observations. Due to\n the other scheduled observations and atmospheric conditions\n we obtained 3 - 40 frames at each night\nwith an exposure time of 5 sec. \nAll images were automatically dark- and flat-field corrected\nas soon as they were exposed.\nFor each corrected image aperture photometry by SExtractor package \n(Bertin $\\&$ \nArnouts 1996) were applied using an aperture of 5 pixels in diameter\nto obtain the instrumental magnitudes. Then these magnitudes were calibrated by comparing all the field stars against USNO A2.0 R-band catalog with a \ntriangle-matching technique.\nBarycentric corrections were made to the times of \neach observation by using JPL DE200 ephemerides prior to the timing analysis \nwith the period determination methods.   \n\n\\section{Results and Discussion}\n\n\\subsection{Pulsation period of {$\\delta$ Scuti } star H254}\n\nWe first attempted to determine the pulsation period of star H254\n(spectral type F0, Harris et al. 1954) \n($\\alpha$=03\\hr 44\\mm 31\\fsec2, $\\delta$=+32\\deg 06\\arcm 22\\farcs1)\nusing our nearly 150 days of\n observational data. Ripepi et al. (2002) identified four frequencies for\n this source by using their eleven days  observations.\n One of these frequencies was at 7.406 $d^{-1}$\n which is typical of {$\\delta$ Scuti } type pulsators. They\nalso reported three more frequencies and explained that these were resulted\nfrom the long term behavior associated with a daily variation of H254\n and partially, with the similar variability in their comparison star H20.\nWe used differential magnitudes which reduces the systematic effects\nsince we are interested in the time series analysis. As a comparison star\nwe chose H89  (see section 3.2) \n($\\alpha$=03\\hr 44\\mm 21\\fsec0, $\\delta$=+32\\deg 07\\arcm 38\\farcs7)\nwhich has a spectral type F8.\nFigure 1 shows ROTSEIIId light curve \n ($\\delta$m$_{R}$=m$_{R}$$^{254}$-m$_{R}$$^{89}$) \nobtained between the nights of MJD 53232 and MJD 53382.\n Period of variation in this light curve was determined by using three separate numerical period searching routines. One is Period98 \n(by Sperl: available at\nwww.astro.univie.ac.at\/$^{\\sim}$dsn\/). The other two are the method of \nScargle (Scargle 1982) and the Clean method (Roberts et al. 1987).\nThese periodograms are essentially discrete Fourier transform of the input \ntime series.\nTo search any periodicity in the  differential light curves, \nwe applied these different period search algorithms mentioned above.\n \\placefigure{fig1}\n For analyzing the periodicity in the light curves periodogram provides an\n approximation to the power spectrum. To be sure about the periodicity\n we applied these different period finding methods.\n \\begin{figure}[h]\n \\epsscale{0.2}\n \\includegraphics[clip=true,scale=0.3,angle=270]{f1.ps}\n \\figcaption{RotseIIId light curve of H254. Error bars on data points \n             are not shown for clarity however estimated errors are\n             of the order $\\sim$0\\fmm02}\\label{fig1}\n \\end{figure}\n\nFigure 2 shows the amplitude and \npower spectra of H254. All of \nthem displays the frequency 7.406 $d^{-1}$.\nThe inset in the first panel shows the window function \nwhich is used to describe the response of data analysis system to a perfect \nsine wave. The peak at one day (and its harmonics) in the periodograms is \na signature of nightly windowing of the sampling frequency.\n \\placefigure{fig2}\n \\begin{figure}[h]\n \\includegraphics[clip=true,scale=0.30,angle=270]{f2.ps}\n \\figcaption{Power spectra for H254. Panel (a): Scargle algoritm, \n             (b): Clean algoritm and (c): Period98. Dotted line\n             on the upper panel represents $3\\sigma$ confidence level. \n             Inset is the spectrum of the window function.}\\label{fig2}\n \\end{figure}\n\nFor the statistics of periodograms we employed the method of Scargle (Scargle\n1982) and evaluated the confidence levels of periodicities. \nWe estimated the noise level of the periodogram by fitting a constant line.\nThe probability of a signal above this level has an exponential\nprobability distribution \n\\[\n  1- P(Z)=  (1- e^{Z})^{N}  \n\\]\nwhich is essentially a $\\chi^{2}$ distribution for two degrees of freedom.\n$Z$ is the power at a given frequency and $N$ is the number of\nfrequencies sampled. For given parameters the confidence level of the signal \nwas found. The confidence level of the signal for the maximum power \nat 7.406 $d^{-1}$ is \n more than $ 5\\sigma $ level signal detection.\n As seen from Figure 2a\nall other detected powers are below the $3\\sigma $ detection level\nwhich indicate that 0.157, 0.283 and 0.931 $d^{-1}$ frequencies\n detected by Ripepi et al. (2002)\nare not present in our light curve. The light curve phased with the frequency\n 7.406 $d^{-1}$ is shown in Figure 3. The amplitude of pulsation is 4.1 mmag\nwhich is comparable with V band amplitude (5.4 mmag) given by Ripepi et al.\n \\placefigure{fig3}\n \\begin{figure}[h]\n \\includegraphics[clip=true,scale=0.3,angle=270]{f3.ps}\n \\figcaption{Light curve of H254 phased with the frequency\n            7.406 d$^{-1}$. }\\label{fig3}\n \\end{figure}\n\n\nThe same period was found using two more different  comparison stars,\n H261 (spectral type F2, \n $\\alpha$=03\\hr 44\\mm 24\\fsec6, $\\delta$=+32\\deg 10\\arcm 14\\farcs4)\n and  H20 (spectral type F8, \n $\\alpha$=03\\hr 43\\arcm 58\\fsec1, $\\delta$=+32\\deg 09\\arcm 47\\farcs5).\nThe amplitude spectra which were obtained\nby using Clean method are shown in Figure 4. The \n peak at the frequency 7.406 $d^{-1}$  corresponding\n to a period of around 3.24 h \n  is seen clearly.\n \\placefigure{fig4}\n \\begin{figure}[h]\n \\includegraphics[clip=true,scale=0.3,angle=270]{f4.ps}\n \\figcaption{Amplitude spectrum of H254 with other set of reference stars \n               (a) H261 (b) H20.}\\label{fig4}\n \\end{figure}\n\n\n\\subsection{Periodic variations in Optical Counterparts of Chandra Sources}\n\nIn this part of the study we searched for the timing properties \nof the optical counterparts of selected Chandra X-Ray sources. \nX-ray images of the cluster IC 348 with the Advanced CCD Imaging Spectrometer \non board the Chandra X-Ray observatory were studied by Preibisch $\\&$ Zinneker\n(2001). They determined the positions\n and count rates of the 215 individual X-ray\n sources. Identification of the\noptical counterparts of the X-ray sources\nwith masses 0.15 and 2 \\hbox{M$_{\\odot}$} were performed for 161 X-ray sources.\n\nThe positions of Chandra sources whose optical counterparts were\nidentified by Preibisch $\\&$ Zinneker (2001) \nwere cross correlated with the positions of\nROTSE objects.\nThe main criteria for the selection is 3\\farcs3\/pixel resolution of the ROTSE\nCCD frames. Hence, to match a known coordinate 3 pixel (10\\arcs) diameter\naperture is used. Secondly, if there is an object closer than 4 pixels \nit is rejected. \nThe exposure time is 5 seconds for each frame. This allows us to\nobserve most of the bright stars of IC 348 without overexposing the frames.\nWith this exposure time stars with  magnitudes between 10 and 14 are \nwell detected and it is also possible to detect stars upto 17th magnitude\ndepending on atmospheric conditions.  \nFor each frame mean FWHM of the point spread function (PSF) is calculated \nfor stars with \n 10 $>$ m$_{R}$ $>$ 14\n and if the mean FWHM $>$ 2 pixel (6\\farcs6) that frame \nis also rejected. \nFigure 5a shows the mean of the magnitude measurement  errors \nassigned to each star in finding their mean magnitudes \nduring the whole observation period.\nFor fainter stars the magnitude determining accuracy decreases.\nThe magnitude errors should be excluded from the measured variations\nin order to obtain the correct intrinsic variabilities. The lower limit for\nthe systematic measurement errors is 0.002 for the brightest star\nin our figure. For magnitudes of stars $\\sim$16 mag, this error\nis about 0.15 mag. As this error increases with increasing magnitude\nthe measurement of intrinsic variability becomes difficult.\n \\placefigure{fig5}\n \\begin{figure}[h]\n \\includegraphics[clip=true,scale=0.3,angle=270]{f5.ps}\n \\figcaption{Upper panel shows mean magnitude errors\n     in calculating the mean magnitudes  for each star in our sample. \n     The mean magnitude of reference stars and their mean errors \n     for each star under consideration are shown in the lower panel.\n     }\\label{fig5}\n \\end{figure}\n\n\\subsubsection{Time Series Analysis}\nTo determine any  time variability in the selected  stars we chose 3 reference \n(comparison) stars (H89, H20, H139).\n To select and check stability of the reference stars, we chose a set of \nstars with  variances less than 0.01 mag over the observing interval. \nPower spectrums of the candidate reference stars were calculated and the ones \nshowing most random power distribution were selected. H89 and H139 from this\nset was used by Cohen et al. (2004) also. \nHence we adopt these stars as the reference stars.\nTwo of these stars have F spectral type and H139  is a G0 star (Luhman\net al. 1998). The average magnitude of the selected\nreference stars were used in the calculation of differential magnitudes.\nFigure 5b  shows the mean magnitudes of the reference stars obtained\nfor each frame against the mean \nmagnitude errors in measuring the magnitude of the reference stars \nfor the data obtained during the observation period  of\nfive months. Each point is the relevant value for the selected stars \nunder investigation. Scatter in the mean magnitude \nvalues of reference stars are due to different number\nof frames used, changing between 800 and 1300, for each selected star.\nThe selection criteria results in a different number of frames for each\nstar. The mean of the reference stars scatter since the\nstar under question and the reference stars  \nare extracted together from each frame.\nDifferential magnitudes of the selected\nobjects are calculated for each frame with the requirements: Selected\nobject and the reference stars are detected with an accuracy of 3 pixels;\nmagnitude error should be less than 0.2 mag; frame should have a PSF\nFWHM $<6\\farcs6$.\nThe mean magnitude of reference stars changes about 0.04 mag\nwhile the deviation from the mean is \nabout 0.001 in mean magnitude measurement errors.\n Each of the three reference stars displays a standard deviation of\nthe order 0.03 magnitude during 5 months of observation period. \n\n\nWe used differential magnitudes in the time series analysis.\nDifferential photometry eliminates the atmospheric and other systematic\neffects over hundred days of observations. These include seeing variations \nin a specific night and between observation days, and also pointing\nvariations of the order of $\\sim0\\fdeg3$ in large FOV ($1\\fdeg8$).\n After the calculation of differential magnitudes \nWe applied the Period98, Clean and Scargle methods \nto obtain the periodograms. The periodograms were calculated for \nthe frequency range between 0 and 20 $d^{-1}$, \nso it was possible to search for periods as short\nas few hours. The time series of each star  was searched\nfor periodicity by using the above mentioned three different period\nsearch methods. \n Most prominent period detected (whose confidence level is greater \nor equal to $5\\sigma$) was given in Table 1 with \nits confidence level which is calculated in the way described in section 3.1. \nThese periods are attributed to the rotation of star with large cool spots \non its surface.  \nThe variance ($\\sigma_{var}$) of the magnitude variations of each star during \nthe observation interval is also shown in the Table. \n\n \\placetable{table1}\n\n\\subsubsection{Periodic Variables}\nWe found 35 stars  as periodic variables.   \nOf the detected 35 periodic variables, 18 stars are new\nperiodic detections.\nThe rest of them whose HMW (Herbst et al. 2000) numbers are given in \ncolumn 3 of Table 1 were studied also by Cohen et al. (2004).\nThe amplitude spectra of 11 newly detected periodic stars obtained\nby applying Clean method\nto the time series data of stars are shown in Figure 6. \nThe rest of them which are stars 3, 20, 51, 71, 73, 143 and 173  are shown in \nFigure 8, 10 and 11.\n \\placefigure{fig6}\n \\begin{figure}[h]\n \\includegraphics[clip=true,scale=0.4,angle=0]{f6.ps}\n \\figcaption{Amplitude spectra of newly detected periodic stars obtained\n             by applying Clean method. Periods corresponding to the detected\n             frequencies for each star  are given in Table 1. }\\label{fig6}\n \\end{figure}\n \\placefigure{fig7}\n \\begin{figure}[h]\n \\includegraphics[clip=true,scale=0.4,angle=0]{f7.ps}\n \\figcaption{Phased light curves of newly detected periodic stars\n             folded at the detected frequencies. Vertical axis is\n             in magnitude units.}\\label{fig7}\n \\end{figure}\nFigure 7 shows phased light curves of the stars shown in Figure 6 at the \ndetected frequencies. Binned phase diagrams are obtained by folding each time\nseries at the detected period. \nThe amplitude of modulation for each star changes between 0.02 and 0.20 \nmagnitude.\n\nWe display the power spectra  of  stars 3 and 20\nin Figure 8, together with their phased light curves. \n Stars 3 and 20 are the samples of stars having shortest and longest \n periods  in our study. \nThe other peaks seen in the top panel are the beat frequencies \nbetween the star's and Earth's rotation periods. In the middle panel \ncleaned dirty spectrum obtained by using Clean method is given.\n \\placefigure{fig8}\n \\begin{figure}[h]\n \\includegraphics[clip=true,scale=0.3,angle=270]{f8.ps}\n \\figcaption{Star 3 and star 20: Stars having the shortest and longest periods in our sample.\n            Upper panel shows the power spectra\n            obtained by Scargle algoritm. Middle panel presents amplitude\n            spectra obtained by Clean algoritm. \n            Lower panel is the phased light curve. }\\label{fig8}\n \\end{figure}\n\nHerbst et al. (2000) indicated that CTTSs were less likely\nto exhibit periodic variations than WTTSs. \nActive accretion can prevent any rotational signature.\nWTTSs are periodic stars. Their cool spots on the surface which are\nstable for several months (Herbst et al. 2000, Cohen et al. 2004)\nallow us to detect the rotation period. These cool spots are expected to \nbe associated with magnetic fields. \nDetection of periodicity could be difficult if the spot pattern\nand places of them change on a timescale of weeks. \nThe periods determined by Cohen et al. and us \nare similar, that is they are similar with a maximum change\nin period  by 1$\\%$ except for star 114.   \nWe observed a period of 15.88 d for this star which\nis greater about a half day compared to the value of Cohen et al.\nThey detected different periods for this star\nin different seasons so they gave an average period for five seasons\nwhich is 16.40 d. There is a period change of 3$\\%$ for this star.\nThis can be related with the chosen Fourier step size which\ngives a maximum error of 0.7 d.  Hence, the stability of rotation periods of\nTT stars over long time scales is confirmed.\nCohen et al. remarks that the longest time that a spot configuration\ncan remain stable enough is between 0.5 and 1 yr.\n \\placefigure{fig9}\n \\begin{figure}[h]\n \\includegraphics[clip=true,scale=0.3,angle=270]{f9.ps}\n \\figcaption{Differential light curves of 6 periodic stars in our\n            sample which were also given by Cohen et al.\n            Numbers for each star refers to HWM catalog.}\\label{fig9}\n \\end{figure}\nIn Figure 9, we plotted the light curves of 6 periodic stars in our\nsample which were also plotted by Cohen et al. \nfor the time interval between 1998 and 2003. \nThese plots make stronger the remarks of Cohen et al. about the change\nof light curve from one season to the other.\n\nThe spectral types of stars that we studied are between A0 and M4.\nFor spectral types earlier than late F type Chandra X-ray\ndetection fraction of the\ncluster is less (Preibisch and Zinnecker 2001). Earlier spectral type\nstars do not show intrinsic X-ray emission. \n Therefore, star 94  did not show rotation \nperiod. For star 187 which is an A0 type star, we found a period of 6.097 d.\nSince we do not expect a chromospheric activity that produce X-ray emission\nfrom this spectral type,\nno rotation period should be observed.\nIt can be explained in the way as Preibisch and Zinnecker (2001) explained;\nthat is this rotation period is due to a very close late type companion\nit is not related with the star itself.\n\n\\subsubsection{CTT Variables}\n\n\n\n\n\n\n \\placefigure{fig10}\n \\begin{figure}[h]\n \\includegraphics[clip=true,scale=0.3,angle=270]{f10.ps}\n \\figcaption{The daily averages of differential light curves of six CTTSs\n             and star 173 which shows large variations in magnitude\n             although classified as WTTS. }\\label{fig10}\n \\end{figure}\n \\placefigure{fig11}\n \\begin{figure}[h]\n \\includegraphics[clip=true,scale=0.42,angle=0]{f11.ps}\n \\figcaption{The amplitude spectra for the \n             CTTSs and star 173 of Figure 10. }\\label{fig11}\n \\end{figure}\nThe daily averages of differential light curves of CTTSs given in Table 1\nare plotted in Figure 10. These are stars 51, 56, 71, 73, 88  and\n143. The star 73 shows a magnitude variation of 0.7 magnitude.\nThe accretion activity is highly variable in time.\nThe continuous activity of this star with its deep minima is seen clearly from\nthe figure. If we think that minima shows the photospheric\nluminosities then the increase in luminosity can be caused by\nthe accretion from a disk around the star. Herbig (1998) classifies this star\nas CTTS (spectral type K0) while Luhmann et al. (1998) measurements of\nH$_\\alpha$ equivalent width indicates a WTTS. \nH$_\\alpha$ emission seems to be time dependent in TT stars (Guenther $\\&$\nEmmerson 1997).\nThe star 143 also\nshows similar variations like star 73, showing a magnitude variations of about\n1.4 mag. Taking typical mass and radius of a K0 star we can calculate the mass accreted on to the star. If increase in luminosity is arising from the accretion from a disk which is comparable to that of gravitational contraction then a\nchange of 0.7 in magnitude corresponds to a mass accretion rate of\n\\.M$ \\sim 10^{19}$ gr\/s. This means that \\.M is $\\sim10^{-7}$\n \\hbox{M$_{\\odot}$}\/yr. \n The other CTTSs are quiet that is variations in their magnitudes\n are small. \n The amplitude  spectra of these stars are given in Figure 11.\nTwo of them (star 51 and 71) show \nrotational periodicities whose periods are given in\nTable 1. Stars 143 and 73 may be in transition phase from\nCTTS to WTTS as Herbst et al. (2000) and Cohen et al. (2004) suggested.\n They also suggested that the deep minima\n seen in the light curves of these stars could be\n caused by occultation events from dust clouds.\nThe maximum powers calculated are above 5$\\sigma$ for these two stars\nat the detected periods of 6.536 d (for star 73) and 32.28 d (for star 143)\nwhich are probably rotation periods. \nAnother star which shows activity in its differential light curve\n like stars 73 and 143\nis star 173 (see Fig.10). This star shows magnitude variations of\n about 1.5 mag. Herbst et al. classifies this star as an active\nnon periodic WTTS since neither previous study gave the strength of \nhydrogen emission line (from which Herbst et al. inferred this line\n was weak). The amplitude spectrum of star 173 (Fig. 11) shows\na periodicity at 22.51 d with a 5$\\sigma$ confidence level.\nThis star may also be thought as CTTS because of its high activity\nsimilar to stars 73 and 143. \nStar 75 whose rotation period was calculated\n as 3.088 d was classified as U (unknown class) by\n Herbst et al. (2000). For this star,\n Luhman et al. (2003) gives the H$_\\alpha$\nequivalent width as 10\\AA. It seems that this star has a\nphase between CTT and WTT. Nevertheless, its light curve is rather quiet;\nit does not show any activity  in its light\n curve as in the case of stars 73, 143 and 173.\n\n\\subsubsection{X-ray Variability and Rotational Periods}\nIn our search we mostly tried to find a period for the optical counterparts\nof the Chandra sources which are classified as WTTSs.\nThe observed X-ray \nemission for WTTSs was explained as coronal emission by\n Preibisch and Zinnecker (2001) and related to the\nstellar rotation.\nAll WTTSs as possible X-ray sources may not have a variability\nduring the time of observation they may be in their spot less\nor changing spot pattern period\nas in the case of stars 77, 103, 115, 148 and 163. To this list\nwe can include also star 17 and 188, since Luhman et al. (2003)\n gives the H$_\\alpha$\nequivalent width smaller than 10\\AA~ for these stars.\n CTTSs have circumstellar accretion disk which could prevent\n the star to show a regular rotation. CTTSs were also detected as X-ray\n sources\n(Preibisch and Zinnecker 2001). Detection frequency among the CTTSs is\n45$\\%$ while among WTTSs it is 73$\\%$. They found no significant\ndifference between the X-ray properties of WTTSs and CTTSs.\n\nPrebisch $\\&$ Zinneker (2002) have shown the light curves (count rates) of\n the sources which shows strong variability during the \nChandra observation. For most of these sources we found rotational\n periods whatever the character of variation of the count rates\n(flare activity, rising or decaying of count rates). \n\n\n \\placefigure{fig12}\n \\begin{figure}[h]\n \\includegraphics[clip=true,scale=0.3,angle=270]{f12.ps}\n \\figcaption{The distribution of rotation periods of sample stars \n          in IC 348 cluster for spectral types earlier than M4}\\label{fig12}\n \\end{figure}\nIn Figure 12, we plot the distribution of rotation\n periods in IC 348 cluster using our\n sample sources for spectral types earlier than M4. \nThe number of stars with slow rotation is less.\nCohen et al. (2004) mentioned about the absence of periods \nshorter than 1 day and deficiency of periods between 4 and 5\ndays. They said these characteristics were also shared\n with the period distributions\nof the Orion Nebula Cluster and Taurus.\n We have two stars whose periodicity is shorter than 1 day.\nThe plot of IC 348 cluster star distribution is similar to Cohen et al.'s\nexcept we have periods shorter than 1 day and longer than 16 days.\n\n\nPlot of rotational period versus spectral type is shown in Figure 13.\nThere is an increase in period towards the later spectral types.\nStars whose spectral types later than K3 have wide range of periods\n between 0.74 and 32 d. However, an overall gradual increase  can not\nbe ruled out.\nWhereas G and early K dwarfs have smaller rotation  periods with a mean \nvalue of $\\sim3.7$ days.\nIn Figure 14, we investigate how TTS's rotation\nis related to the chromospheric and coronal activity.\nX-ray luminosities of the sample stars given by \\citet{pre02} were plotted\nagainst the rotational period.\nBouvier (1990) proposed that the correlation between X-ray fluxes and\n rotational periods of TTSs was caused by a solar type dynamo which is \nresponsible for the chromosperic and coronal activity of stars as it is \nin active dwarfs.\nDespite the large scatter in the data, there is a trend toward decreasing\nX-ray luminosity as the rotation period increases.\nOn the other hand stars with periods $<$4 d have \nan average X-ray luminosity of $\\sim 2\\times10^{30}$ erg\/sec \nwith a large scatter.\nAs the stars rotate faster their chromospheric and coronal activity increases.\nRotation seems to be an important parameter which influences the level of\nX-ray emission of stars.\nWe note that WTTSs and CTTSs exhibit similar X-ray luminosities \nat any rotational period.\nWe conclude that X-ray luminosities of TTSs in IC 348 cluster depend on\n rotation.\n\n\n \\placefigure{fig13}\n \\begin{figure}[h]\n \\includegraphics[clip=true,scale=0.3,angle=270]{f13.ps}\n \\figcaption{The distribution of rotation periods of sample stars\n          in IC 348 cluster. \n           Open circles denote WTTSs and triangles are CTTSs.\n           }\\label{fig13}\n \\end{figure}\n\n \\placefigure{fig14}\n \\begin{figure}[h]\n \\includegraphics[clip=true,scale=0.3,angle=270]{f14.ps}\n \\figcaption{The distribution of X-ray luminosity  of sample stars\n          in IC 348 cluster as a function of rotational periods.\n          Open circles denote WTTSs and triangles are CTTSs.}\\label{fig14}\n \\end{figure}\n\n\\section{Summary}\n\nThe main results of our analysis of the ROTSE observations of IC 348 \ncluster can be summarized as follows:\n\nWe have 5 months of continuous data of this cluster.\nIn the time series analysis of the stars for the frequency range\n between 0 and 20 $d^{-1}$ we did not find any periodicity shorter than \n0.7 d. Only for the star H254 we confirmed the  \n{$\\delta$ Scuti } pulsation period of 3.24 hr. The other frequencies detected\n0.157, 0.283 and 0.931 $d^{-1}$\nby Ripepi et al. (2002) for  H254 were not present in our light curve.\n\nWe found 35 stars as rotationally periodic stars whose rotation\nperiods change between 0.74 and 32.3 d. 18 of them were  newly \ndetected periodic stars. 8 of the 18 stars (stars  20, 62, 73, 122,\n143, 153, 159, 173) were also studied by \nCohen et al. but they did not give any period for\nthese stars. That can be due to the unstable spot patterns\nduring their observation periods. Perhaps the observation duration was\nnot enough to determine the periods.\nCohen et al. noted that stars may remain spotted but the spot pattern\nevolves such that a period can not be determined over 6 consecutive months\nof observation. Since we detected the periods of these stars it is\nprobably related with the number of data points used in the time series. \nIf the size of the spot is small it can be difficult to detect the period. \n\nMost of the stars whose periods were detected were WTTSs.\nThe periods of non accreting WTTSs are easily detected.\nThere were 7 non periodic WTTSs in our analysis. This may be due\nto a changing spot pattern or spot less period of the star\nduring the observation.\nFor one of them (star 103) Cohen et al. gives a period of\n2.237 d which is an average over all of five seasons\nthey studied. They could not find periodicity for two seasons \nfor this star.\nThe number of CTTSs that we study is less than WTTSs.\nFor the 4 CTTSs (51, 71, 73, 143) we detected rotation period.\nIt would not be possible to detect the periods if the disk\nprevents the detection of rotational variability.\nPrebisch $\\&$ Zinneker (2002) noted that M type stars without any\ncircumstellar material can show H$_\\alpha$ emission.\nThey may not have an accretion disk.\nH$_\\alpha$ emission can also be time dependent in TT stars.\nThese 4 stars may be seen as CTTS at the time of measurement of\nH$_\\alpha$ emission, but at another time H$_\\alpha$ emission\n may be less and they may appear as WTTS. \n\nThe rotational periods found in this study are similar with those of \nCohen et al. with a maximum change of 1$\\%$ in period. \nSmall changes in the rotational periods indicate a rigid rotation.\nRotation periods seems to be stable on a timescale of \n$\\sim$6 yr in this cluster \nwhen evaluated together with the results of Cohen et al.\n\n\nWe found an inverse correlation between X-ray luminosity and the rotational\nperiod in our sample of late type TTSs.\nX-ray luminosity decreases as the  stars\nrotate slower. WTTSs and CTTSs bahave similar in X-ray activity\nat any rotational period. The dispersion in rotational periods at a given\nspectral type results in a  dispersion in X-ray luminosity.\n\n\\acknowledgments\n\nWe thank the referee Prof. Kevin Luhman, for a careful reading and \nvaluable comments.\nThis project utilizes data obtained by the Robotic Optical Transient Search\nExperiment.  ROTSE is a collaboration of Lawrence Livermore National Lab, \nLos Alamos National Lab, and the University of Michigan\n(www.umich.edu\/$\\sim$rotse).\nWe thank the Turkish National Observatory of TUB\\.ITAK\nfor running the optical facilities. This research has made use of the \nSIMBAD database, operated at CDS, Strasbourg, France. \nSpecial thanks to Tuncay \\\"Oz{\\i}\\c{s}{\\i}k from TUG \nwho keeps hands on ROTSEIIId.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nChepoi, Estellon and Vax\\`{e}s showed there is a constant $\\rho$ such\nthat any planar graph of diameter at most $2R$ has a subset of at most\n$\\rho$ vertices such that every vertex in the graph is within distance\n$R$ of that subset~\\cite{planarballcover}. Since this can be viewed as\nshowing that there is a constant-sized set cover in the set system of\nballs of radius $R$, we refer to this property as the {\\em ball-cover}\nproperty.  Graphs having constant-sized ball covers admit interval\nrouting schemes with dilation $\\frac{3}{2} \\times \\text{diameter}$ and\ncompactness $O(1)$ where dilation measures the indirectness of the\nrouting scheme and compactness measures the size of the routing\ntable~\\cite{GPRS01}.  We believe the ball-cover property is an\ninherently interesting property.  Graphs having this property could\ndefine an interesting class of graphs and perhaps could have broader\nutility than previously realized.\n\nWe generalize the class of graphs having the ball-cover property to\nthose graph families that can be embedded on a surface of fixed genus\nafter the removal of a constant number of vertices (the {\\em apices});\nthe number of balls required depends only on the genus of the surface\n(either orientable or non-orientable) and the number of apices.  Since\ngraphs of bounded treewidth are also known to have the ball-cover\nproperty~\\cite{GPRS01} by way of the Graph Minor Structure Theorem,\nour result is a significant step toward proving that\nfixed-minor-excluded graphs also have the ball-cover property.  \nWe discuss this more in\nSection~\\ref{sec:apex}.  We start by sketching the proof for the\nplanar case as we use a similar, but more general, tool set here.\n\n\n\\subsection{A sketch of the proof of the ball-cover property for planar graphs}\n\nThroughout, graphs are simple, undirected and unweighted.  Let $B(x)$\nbe the set of all vertices that are within distance $R$ of vertex $x$\nin graph $G$; this is the {\\em ball centered at $x$}. Let ${\\cal B}(G)\n= \\{B(x)\\ : \\ x \\in V(G)\\}$; this is the {\\em ball system} of $G$. We\nsay that ${\\cal B}' \\subset {\\cal B}$ covers $G$ if ${\\cal B}'$ is a\nset cover of $V(G)$.\n\nThe dual of a set system $\\cal S$ with ground set $U$ is defined as\nfollows: the ground set of the dual set system is $\\cal S$ and for\nevery element $x \\in U$, the dual system has a set representing the\nsets of $\\cal S$ containing $x$, i.e., $X = \\{ S\\ : \\ S \\in {\\cal S},\nx \\in S\\}$.  It is easy to see:\n\\begin{observation} \\label{obs:dual}\n  The dual set system of ${\\cal B}(G)$ is ${\\cal B}(G)$.\n\\end{observation}\nSince a hitting set of $\\cal S$ (a subset of the ground set that\ncontains an element in every set) is a set cover of the dual set\nsystem of $\\cal S$, we likewise have that the centers of a subset of\nballs covering $G$ is a hitting set for the ball system.  A hitting\nset of ${\\cal B}(G)$ is exactly a subset of vertices within which\nevery other vertex is distance $R$.\n\nMatou\\u{s}ek gives a characterization of set systems that have small\nhitting sets~\\cite{Matousek04} in terms of the set system's {\\em\n  fractional-Helly} or {\\em $(p,q)$-property} and the dual set\nsystem's {\\em VC-dimension}.\n\n\\subsubsection*{VC-dimension} A set system $\\cal S$ {\\em\n  shatters} a set $X$ if for every subset $Y$ of $X$ there is a set $S\n\\in {\\cal S}$ such that $S \\cap X = Y$. The Vapnik-Chervonenkis\ndimension or VC-dimension of $\\cal S$ is the maximum size of a set\nthat $\\cal S$ can shatter~\\cite{VC71}.  Chepoi et~al. remark that\nthe VC-dimension of the ball system of a graph excluding $K_{r+1}$ as\na minor is at most $r$~\\cite{planarballcover}.  This gives us:\n\\begin{lemma}\\label{lem:vc-dim}\nThe VC-dimension of ball system of a graph excluding $H$ as a\nminor is at most $|H|-1$.\n\\end{lemma}\nRecall that a minor of a graph $G$ is a graph that is obtained from\n$G$ by edge contractions and deletions; a forbidden or excluded minor\nis a graph that {\\em cannot} be obtained this way.  It follows from\nObservation~\\ref{obs:dual} that the dual of the ball system of a graph\nexcluding $K_{r+1}$ as a minor also has VC-dimension at most $r$.\n\n\\subsubsection*{Fractional Helly theorems} If a set system is such that\nevery $d$ sets has a point in common, then the set system is said to\nhave Helly order $d$. A {\\em Helly theorem} is one that shows that\ncertain set systems of Helly order $d$ have a non-empty intersection.\nThe first such theorem was given for the Euclidean plane: if a family\nof convex sets has a nonempty intersection for every triple of sets,\nthen the whole family has a nonempty intersection~\\cite{Helly23}.  A\nset system has {\\em fractional} Helly order $(p,q)$, or {\\em has the\n  $(p,q)$-property}, if among every $p$ sets some $q$ have a point in\ncommon. Matou\\u{s}ek gave the following fractional Helly theorem:\n\\begin{theorem}[Fractional Helly Theorem~\\cite{Matousek04}]\\label{thm:pq}\n  Let $\\cal Q$ be a set system having the $(p,q)$-property (for\n  $p \\ge q$) and whose dual set system has VC-dimension $q-1$.  Then there\n  is a constant $\\rho$ such that $\\cal Q$ has a hitting set of size at\n  most $\\rho$.\n\\end{theorem}\n\nGiven Lemma~\\ref{lem:vc-dim}, one could therefore show that, for a\nfixed minor $H$, $H$-minor free graphs have the ball-cover property\nby showing that the corresponding ball system has fractional Helly order $(p,\n|H|)$ for some fixed $p \\ge |H|$.  Chepoi et~al.\\\ndo just this for planar graphs.  Starting with $p$ vertices, they consider the pairwise\nshortest paths between these vertices; each shortest path contains a\nvertex that is contained by the balls centered on the paths'\nendpoints.  Viewing these shortest paths as edges of a complete graph\nand drawn on the plane (as inherited from a drawing of the original\ngraph), they invoke a result showing that such a drawing of $K_p$, for $p$ sufficiently large, must contain at least 7 pairwise\ncrossing edges.  The 7 pairwise crossing shortest paths then witness a\npoint in common to 5 of the balls.  We use this idea at the heart of\nour proof for surface-embedded graphs.\n\n\\subsection{Surface-embedded graphs}\n\nWe start by extending this result to graphs embedded on more general\nsurfaces. We first give some definitions.\n\n A $2$-manifold (or surface) $S$ is a Hausdorff space in\nwhich every point has a neighborhood homeomorphic to the Euclidean\nplane or the closed half plane.  A cycle in a surface is a continuous\nfunction from $S^1$ to the surface; the cycle is called simple if the\nmap is injective. A simple cycle $\\gamma$ is separating if $S\n\\backslash \\gamma$ is not connected; see Figure~\\ref{fig:homologous}.  The genus $g$ of a surface $S$\nis the maximum number of pairwise disjoint non-separating cycles $\\gamma_1,\n\\gamma_2, \\ldots, \\gamma_g$ such that $S \\setminus (\\gamma_1 \\cup \\cdots \\cup \\gamma_g)$ is\nconnected.  \nNote that cutting a surface along a non-separating cycle reduces the genus by 1; this is a common\nalgorithmic technique for reducing the complexity of a surface.\nA surface is non-orientable if and only if it contains a\nsubspace homeomorphic to the M\\\"{o}bius band and is \notherwise orientable.\n\nAn embedding of a graph $G=(V,E)$ on a surface $S$ is a drawing of $G$\non $S$, such that vertices are mapped to distinct points in $S$ and\nedges are mapped to \\emph{internally} disjoint simple paths.  A face\nof an embedding is a maximal connected subset of $S$ that does not\nintersect the image of $G$.  An embedding is cellular if all of its\nfaces are homeomorphic to a topological open disc.  We say that $G$ is\na graph of (orientable or non-orientable) genus $g$ if $G$ has\na cellular embedding on a surface of (orientable or non-orientable)\ngenus $g$.\n\nWe will briefly use the notion of $\\mathbb{Z}_2$-homology in this paper and so include a \nbrief description for completeness; we refer the reader to a topology text for full \ndetails~\\cite{h-at-02,m-t-00}.  A homology cycle is a \nlinear combination of oriented cycles with coefficients from a ring\n$R$; when $R=\\mathbb{Z}_2$, these homology cycles are even-degree subgraphs of $G$.  \nA boundary subgraph is the boundary of a union of faces of $G$.  \nTwo subgraphs are homologous if their symmetric difference is a boundary \nsubgraph, or, more intuitively, if\nthey can be deformed to each other (where the deformation may include\nmerging intersection cycles or splitting at self-intersections or\ndeleting trivial separating cycles); see Figure~\\ref{fig:homologous} for an example.\nBoundary cycles are null-homologous, and since every separating cycle is a boundary cycle, we can view separating cycles as the identity element for homology classes.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3in]{homologous1.pdf}\n\\includegraphics[width=3in]{splitting.pdf}\n\\end{center}\n\\caption{Left: An example of homologous cycles: the single dashed red non-separating cycle (above on left) is $\\mathbb{Z}_2$-homologous to two solid blue cycles. Right: A null-homologous separating cycle.}\n\\label{fig:homologous}\n\\end{figure}\n\n\\subsection{Our contribution}\n\nThe bulk of this paper focusses on showing that a graph of genus $g$\nhas the ball-cover property by showing that its ball system has the\n$(p_g,q_g)$-property for numbers $p_g$ and $q_g$ that depend only on\n$g$ (Section~\\ref{sec:pq}). Since $K_n$ has orientable genus $\\lceil \\frac{1}{12}(n-3)(n-4)\n\\rceil$ and non-orientable genus $\\lceil \\frac{1}{6}(n-3)(n-4)\n\\rceil$\\cite{ringel-youngs68}, we set $q_g = c\\cdot g^2$ (where $c$\ndepends only on whether the surface in question is orientable).  Then, since a\ngraph of genus at most $g$ excludes $K_{q_g}$ as a minor, the\nVC-dimension for a graph of genus at most $g$ is at most $q_g-1$.  By\nObservation~\\ref{obs:dual}, Lemma~\\ref{lem:vc-dim} and the Fractional\nHelly Theorem, we will get:\n\\begin{theorem} \\label{thm:main}\n  There exists a constant $\\rho_g$ (depending only on $g$) such that\n  any graph of genus at most $g$ and diameter at most $2R$ can be\n  covered by at most $\\rho_g$ balls of radius $R$.\n\\end{theorem}\nWe show that the same holds if the graph additionally has a fixed\nnumber of apices and discuss how one might generalize to\nfixed-minor-excluded graph families in Section~\\ref{sec:apex}.\n\nIn order to prove that the ball system for a genus-$g$ graph has the\n$(p_g,q_g)$-property, we show that there is a small set of edges of a\nsurface-embedded graph whose removal leaves a planar graph\n(Section~\\ref{sec:sep}) and give bounds on the number of edges in a\ngraph drawn on a surface of fixed genus having a limit on the number\nof crossings (Section~\\ref{sec:cross}).  The former result can be used\nto generalize an edge-separator result for planar graphs due to Gazit\nand Miller~\\cite{GaMi90}.  Both these\nresults are likely of more general interest.  We give background on\nthese problems in their relevant sections.  \n\nThe takeaway from these generalizations will allow us to argue that\nany topological drawing of $K_n$ on a surface of orientable or\nnon-orientable genus $g$ must have a large subset of edges that\npairwise cross.\nIn Section~\\ref{sec:cross}, we will formally define what constitutes a\ntopological drawing on a surface of genus $g$ and prove this theorem.\n\n\n\\section{A norm-sized, planarizing edge set for surface-embedded graphs} \\label{sec:sep}\n\nIn this section, we show there is a small set of edges in a surface-embedded graph whose removal leaves a planar graph.  We start by bounding the size of a non-separating cycle:\n\\begin{theorem}\\label{thm:cycle}\n  The shortest non-separating cycle of a graph $G$ embedded on a surface has length at most ${1\\over 2}||G||_f$.\n\\end{theorem}\nwhere \n\\[\n||G||_f = \\sqrt{\\sum_{f \\in {\\cal F}} |f|^2}\n\\]\nis the {\\em face-norm} of $G$ and ${\\cal F}$ is the set of faces of\n$G$.  We use a sequence of $g$ non-separating cycles to {\\em\n  planarize} $G$.  The face-norm was used by Gazit and Miller to\ntighten the bound on the size of edge-separators for planar\ngraphs~\\cite{GaMi90}.  Theorem~\\ref{thm:cycle} implies an $O(g\n||G||_f)$-sized edge separator for genus-$g$ graphs.  We discuss some\nopen problems in this vein at the end of the paper.\n\nLet $G$ be a graph with a cellular embedding on a surface of genus $g$\n(either orientable or not).  We start with a shortest non-separating\ncycle $C$ and generate an ordered family of disjoint cycle sets $\\cal\nC$ each of which is homologous to $C$.  We use this family to build\nanother non-separating cycle $C'$ formed by one vertex from each set\nin $\\cal C$.  Since $C$ is shortest, $C'$ acts a witness giving a\nlower bound on $|\\cal C|$.  Overall, this gives a lower bound on the\nnumber of edges in $\\cal C$, and so an upper bound on $|C|$.\n  \nWe appeal to a combinatorial embedding of the graph which gives, for\neach vertex $v$, a clockwise ordering of the edges incident to $v$ as\nthey are embedded around $v$~\\cite{Edmonds60,Youngs63}.  We note that any such embedding can be\nmaintained under operations such as contraction, deleting, or cutting\nalong a cycle, via appropriate unions, deletions, or duplications of\nthe vertex lists which maintain the clockwise orderings; full details are described by Mohar and Thomassen~\\cite{mt-gs-01-ch4}.\n\nIn the following $\\partial f$ denotes the boundary of face $f$.\n\n\\begin{lemma} \\label{lem:homol}\n  Let $G$ be a graph with a cellular embedding on a surface $\\cal S$, either orientable or non-orientable.  Let $\\cal F$ be\n  a set faces of $G$.  We can add a set $L$ of edges to $G$ such that\n  \\begin{itemize}\n  \\item $L$ can be incorporated into the embedding of $G$ in a noncrossing way.\n  \\item The endpoints of $L$ are the set of vertices at distance one from the\n    boundaries of $\\cal F$.\n  \\item $L$ decomposes into a set of cycles that is homologous to the boundaries of\n    $\\cal F$.\n  \\end{itemize}\n\\end{lemma}\n\n\n\\begin{proof}\n\n  For a face $f \\in {\\cal F}$, let $\\partial f$ denote the cycle in\n  $G$ giving $f$'s boundary, taken in clockwise order.  Let $X$ be the\n  set of vertices at distance 1 from $\\cal F$ in $G$.  If $f, g \\in\n  {\\cal F}$ are adjacent in $G$ (that is, there is an edge $uv$ such\n  that $u \\in \\partial f$ and $v \\in \\partial g$ or $f$ and $g$ share\n  a vertex $x$), then the vertices at distance 1 from $\\partial f$\n  interferes with $\\partial g$.  To avoid this, we merge adjacent\n  faces.  If $f$ and $g$ share a vertex $x$, we cut open the graph at $x$, merging the interiors of $f$ and $g$ and creating two copies of $x$, both on the boundary of the newly created face.  If $f$ and $g$ are connected by an edge $uv$, we cut open the graph along $uv$, merge the interior of\n  $f$ with that of $g$ resulting in face $h$.  The edge $uv$ is\n  duplicated and both copies appear in $\\partial h$.  We repeat this\n  operation {\\em minimally} until the distance between every pair of faces is at least\n  2: that is, performing a sequence of such operations will guarantee that the interior of the resulting faces are homeomorphic to a disk. \n (Note that on a non-orientable surface, this minimality avoids the possibility that the union of neighboring faces spans a M\\\"obuis strip, and so the interior remains a topological disk.) \n   Let ${\\cal F}'$ be the resulting set of faces and $G'$ the\n  resulting graph.  Note that the boundaries of $\\cal F$ are\n  $\\mathbb{Z}_2$-homologous to the boundaries of ${\\cal F}'$, since\n  the introduction of $uv$ twice cancels under $\\mathbb{Z}_2$\n  homology.  Note further that interior of each face in ${\\cal F}'$ is\n  homologous to a disk and thus the boundaries are contractible, and\n  the set of vertices at distance one from ${\\cal F}'$ is still $X$,\n  the set of vertices at distance 1 from ${\\cal F}$.\n\n  Let $G''$ be the graph obtained by contracting the boundaries of the\n  faces of ${\\cal F}'$.  Let $F'$ be the vertices resulting from these\n  contractions.  Note again that the set of vertices at distance 1 from\n  $F'$ in $G''$ is still $X$, since each vertex at distance 1 from $F'$ must also be within distance 1 of some vertex in $\\partial F$, and vice versa.\n\nWe will build a cycle that is homologous to each face in ${\\cal F}'$ whose vertices are among $X$.  Since the faces in ${\\cal F}'$ are at distance at least two from each other, the cycles we construct will not interact with each other.\n\n\n \\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=3in]{leveledges2}\n    \\caption{Cycles which are connected by an edge are merged into a single face (shaded above, right), and level edges (shown dashed above) are embedded so that the boundary of the face, incident edges and new edge bounds a topological disk (shaded above, left).}\n    \\label{fig:cycles}\n  \\end{figure}\n\nSubdivide every self-loop $\\ell$ adjacent to a vertex\nin ${\\cal F}'$ into two edges  with a vertex $v_\\ell$.  Let $G'''$ be the resulting graph.  The set of vertices at distance 1 is now $X'$, which consists of vertices from $X$ and vertices which came from loop subdivisions.\n\nFor each vertex $f \\in F'$, consider the cyclic clockwise ordering of the\nedges incident to the vertex corresponding to $f$ in the embedding of $G'''$.  For every two\nconsecutive edges $fu$ and $fv$ in this order we introduce the edge $uv$ and call it\na {\\em level edge}.  Edge $uv$ can be embedded to be arbitrarily close to $fu$ followed by $fv$; on the original surface, this corresponds to a path following the edge from $u$ to the face $f$, followed by a (possibly empty) portion of the face boundary $\\partial f$, followed by the edge from $f$ to $v$; see Figure~\\ref{fig:cycles}.\nLet $L$ be the set of all such\nedges.   Since each such edge can be embedded as described to follow two adjacent edges in the clockwise ordering around the vertex $f$, $G''' \\cup L$ can be embedded in a non-crossing way.  Note that self-loops and parallel edges may be introduced this way, e.g.\\ when a vertex $f \\in F$ has degree 1 or 2, respectively.  See Figure~\\ref{fig:loopleveledges}.\n\nThe level edges corresponding to $f$ inherit a cyclic ordering from\nthe ordering of the edges adjacent to $f$.  That is, $uv$ and $vw$ are\nconsecutive in this ordering if $fu,fv,fw$ are consecutive in the\nordering of edges adjacent to $f$. Further, given how we have embedded $uv$, we know that the cycle $\\partial f$ union the edges $fu, uv, fv$ bounds a topological disk.\nThis implies a partitioning of $L$ into a set of\ncycles ${\\cal C}$ that is homologous to the boundaries of ${\\cal F}'$: simply replace each portion of a face $\\partial f$ with the path $fu, uv, fv$.  Since we are (in $\\mathbb{Z}_2$ homology sense) adding a set of disks to a cycle, each new cycle is homologous to the original.  This proves the second and third implications of Lemma~\\ref{lem:homol}.\n\nHowever, the endpoints of $L$ are not necessarily vertices of $G$,\nsince they include the subdividing vertices.  Refer to Figure~\\ref{fig:loopleveledges}.  Consider such a vertex\n$v_\\ell \\in X'$ which was used to subdivide self-loop $\\ell$.  Merge\nany two consecutive edges $uv_\\ell$, $v_\\ell w$, creating edge $uw$\nand minimally modify the embedding so that $uw$ does not intersect\n$\\ell$.  This maintains the second and third implications.\nIf there are parallel loops (either on an oriented or non-oriented surface), the connecting level edges consist of bigons between loop vertices; these bigons are null-homologous and hence can be disregarded. The set of\nlevel edges may also have included a self-loop centered at a subdividing\nvertex, the new ``edge'' will no longer have any endpoints.  This\n``edge'' must bound a topological disk, since, if we introduced a\nlevel edge centered at $v_\\ell$, $\\ell$ must have bounded a face in\n$G''$.  Therefore, we can remove this ``edge'' while maintaining the same homology type for our set of cycles.  We let $L'$ be the modified and remaining edges.  These are the edges satisfying the three implications of Lemma~\\ref{lem:homol}. \\qed\n\n\n\n\n\n\n\n\n \\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1.5in]{loopleveledges1}\n    \\includegraphics[width=1.5in]{loopleveledges2}\n    \\caption{A face $f$ (shaded) and its incident edges (solid) with added subdividing vertices (hollow).  Left: the level edges $L$ (dashed) added to $G'''$.  Right: the level edges after connections to the subdividing vertices are removed.  Note that the outer endpoint-less ``edge'' is not included in $L'$.}\n    \\label{fig:loopleveledges}\n  \\end{figure}\n\n\n\n\\end{proof}\n\n\\subsection{Short non-separating cycles}\n\nWe are now ready to prove Theorem~\\ref{thm:cycle}.\n\nLet $C$ be the shortest non-separating cycle of $G$.  Cut open the\nsurface and graph along $C$, duplicating $C$ into copies $C_0$ and\n$C_0'$; let $G_0$ be the cut open graph.  Glue a disk onto each hole\nleft from cutting open the graph.  $C_0$ and $C_0'$ are now the\nboundaries of faces.\n\nLet $V_i$ be the set of vertices in $G_0$ that are at distance $i$\nfrom $C_0$ and let $s$ be the smallest index such that $V_s \\cap\nV(C_0') \\ne \\emptyset$.  We define sets of cycles $C_i$ in a graph\n$G_i$, $i = 0 \\ldots, s$, starting with $C_0$, inductively as follows:\nGiven the set of cycles $C_{i-1}$ that are the boundaries of faces\n(and starting with $C_0$ as our initial cycle), we define $C_i$ to be\nthe homologous set of cycles going through $V_i$ as guaranteed by\nLemma~\\ref{lem:homol}.  We remove the edges and vertices of $C_{i-1}$\nand the edges adjacent to $C_{i-1}$ to make $C_i$ the boundaries of\nfaces.\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n  For any chord $uv$ of a face $f$, let $P_{uv}$ be the shortest\n  $u$-to-$v$ path along the boundary of $f$ and let $\\ell(uv) =\n  |P_{uv}|$.  Gazit and Miller~\\cite{GaMi90} show that for a face $f$\n  and a set of pairwise non-crossing chords $H$ across $f$, $\\ell(H) \\le \\frac{1}{8}|f|^2$.\n  Since the edges of $\\cup_{i = 1}^s C_i$ are chords of the faces of $G$, we get\n  \\begin{equation}\n    \\label{eq:level-wts}\n    \\sum_{i=0}^s \\ell(C_i) \\le {1\\over 8} (||G||_f)^2\n  \\end{equation}\n\n  By construction $C_i$ is homologous to $C_0$ and so to $C$.  Let\n  $\\bar C_i$ be the set of cycles obtained from $C_i$ by replacing\n  each edge $uv \\in C_i$ with $P_{uv}$.  We get $|\\bar C_i| =\n  \\ell(C_i)$.  Since $\\bar C_i$ is homologous to $C$, $\\bar C_i$ must contain a non-separating cycle $S$.  Since $C$ is the shortest non-separating cycle,\n  \\[\n  |\\bar C_i| \\ge |S| \\ge |C|\n  \\]\n\n  \\begin{figure}[hb]\n    \\centering\n    \\includegraphics{abc.pdf}\n    \\caption{$C$ is the shortest non-separating cycle, $A \\cup B$ is another non-separating cycle.}\n    \\label{fig:ABC}\n  \\end{figure}\n\n  Let $B$ be the shortest path from $C_0$ to $C_0'$.  Let $A$ be the\n  shortest subpath of $C$ that connects $B$'s endpoints.  $A \\cup B$\n  is a non-separating cycle.  See Figure~\\ref{fig:ABC}  Since $|B| = s$ and $|A \\cup B| \\ge\n  |C|$, $s \\ge |C|\/2$.  We have:\n  \\[\n  {1\\over 8}(||G||_f)^2 \\ge \\sum_{i=1}^s\\ell(C_i) =  \\sum_{i=1}^s |\\bar C_i| \\ge |C|^2\/2 \n  \\]\n  Rearranging gives Theorem~\\ref{thm:cycle}.\n\n\\subsection{Planarizing sets}\n\nRepeatedly cutting along non-separating cycles allows us to reduce a surface-embedded graph to a planar graph, while only reducing the face norm:\n\n\\begin{lemma}\\label{lem:cut-n-contract}\n  Let $G$ be an embedded graph and let $C$ be a non-separating cycle.\n  Cutting open the graph along $C$ and then contracting each resulting\n  copy of $C$ results in a graph $G'$ such that $||G'||_f <\n  ||G||_f$.\n\\end{lemma}\n\n\\begin{proof}\n  Let $\\cal F$ be the set of faces of $G$ and let ${\\cal F}_C$ be the\n  set of faces of $G$ that have a bounding edge in $C$.  Cutting along\n  a non-separating cycle $C$ of a graph $G$ embedded on surface $\\cal\n  S$ introduces two holes, each bounded by a copy of $C$.  Contracting\n  each hole and each bounding copy of $C$ results in a graph $G'$ with\n  face set ${\\cal F}'$.  Every face in ${\\cal F}$ maps to a face in\n  ${\\cal F}'$ such that the faces in ${\\cal F} \\setminus {\\cal F}_C$\n  are the same size as their image in ${\\cal F}'$ and the faces in\n  ${\\cal F}_C$ are strictly larger than their counterparts ${\\cal F}'_C$ in ${\\cal\n    F}'$ giving:\n  \\[\n  ||G'||_f = \\sqrt{\\sum_{f \\in {\\cal F}'} |f|^2 }\n = \\sqrt{ \\sum_{f \\in {\\cal F}'\\setminus{\\cal F}'_C} |f|^2 +\\sum_{f \\in {\\cal F}'_C} |f|^2 } < \\sqrt{ \\sum_{f \\in {\\cal F}\\setminus{\\cal F}_C} |f|^2 +\\sum_{f \\in {\\cal F}_C} |f|^2 } = \\sqrt{\\sum_{f \\in {\\cal F}} |f|^2 }= ||G||_f \\] \\qed\n\\end{proof}\n\nCutting along a non-separating cycle $C$ of a graph $G$ embedded on\nsurface $\\cal S$ reduces the genus of the surface by one and\nintroduces two holes, each bounded by a copy of $C$.\nLemma~\\ref{lem:cut-n-contract} shows that if we contract the two\ncopies of $C$ (and the corresponding holes), we only reduce the\nface-norm of the graph.  We can repeat this cut-and-contract procedure\n$g$ times, each time we find a non-separating cycle of length at most\n${1\\over 2}||G||_f$, at which point the surface is a sphere and the\nfinal graph $G'$ is planar.  Of course, applying this method to the\ndual $G^*$ of the graph, results in a set of {\\em planarizing} edges\nwhose size is measured in terms of the vertex-norm\n\\[\n||G||_\\delta = \\sqrt{\\sum_{v \\in V} \\delta(v)^2}\n\\]\nof $G$ where $\\delta(v)$ is the degree of vertex $v$.  Recall that the\ndual of a plane graph is given by a vertex for every face of the primal\ngraph, with dual vertices connected when the corresponding primal\nfaces are adjacent.  By duality, the degree of a vertex is the size of\nthe face in the dual corresponding to the vertex.  We get:\n\\begin{lemma}\\label{lem:planarize}\n  There is a set of $\\frac{g}{2}||G||_\\delta$ edges of a genus-$g$\n  graph whose removal leaves a planar graph.\n\\end{lemma}\n\n\n\n\\section{Pairwise-crossing number of surfaces}\\label{sec:cross}\n\nThere are many measures of how close a graph is to being planar.  One measure is the crossing number which is the minimum number of edge crossings in a planar, topological drawing of the graph~\\cite{Turan77}.  A drawing is {\\em topological} if vertices map to distinct points and edges map to simple Jordan arcs connecting the points their endpoints such that (i) no arc passes through a vertex different from its endpoints, (ii) no two arcs meet in more than one point, and (iii) no three arcs share a common interior point.  Formally the crossing number of a fixed drawing is number of interior points that are shared by two arcs.  The restriction to topological drawings does not increase the crossing number of a graph, see e.g.~\\cite{Felsner04}.  Rather than planar drawings, we are interested in drawings on surfaces of genus $g$ and so will refer to {\\em surface topological drawings}.  This number has been studied by Shahrokhi, Sz\\'{e}kely, S\\'{y}kora and Vrt'o, who give upper and lower bounds on the crossing number of complete graphs drawn on compact 2-manifolds~\\cite{SSSV94,SSSV96}; more specific bounds are also known for surfaces such as the torus~\\cite{Guy1968376}.\n\nWe first use the crossing number of a particular drawing of a graph to give bounds on the size of a set of edges whose removal results in  a topological drawing in the plane.\n\n\\begin{lemma}\\label{lem:topo-planarize}\nA graph $G$ admitting  a topological drawing on a surface $\\cal S$ of genus $g$ with $\\chi$ crossings has a subset of at most \n\\[ {g \\over 2}\\sqrt{16\\chi+||G||_\\delta^2}\\] edges whose removal\nleaves a graph whose inherited drawing is a planar topological\ndrawing.\n\\end{lemma}\n\n\\begin{proof}Let $H$ be the graph embedded on $\\cal S$ obtained\nfrom $G$ by introducing a vertex at each crossing.  Since the drawing is topological, each of these new vertices has degree 4.  We have that $||H||_\\delta^2 = \\sum_{v\\in H} \\delta_H(v)^2 = 16\\chi+\\sum_{v\\in G} \\delta_G(v)^2$.  By Lemma~\\ref{lem:planarize}, $H$ has a planarizing edge set $S_H$ with at most ${g\\over 2}||H||_\\delta$ edges.  Let $S_G$ be the set of edges of $G$ from which $S_H$ are generated.  Since $|S_G| \\le |S_H|$, the lemma follows.\n\\hfill \\qed \\end{proof}\n\nAnother class of graphs that is close to being planar are the class of\n{\\em $k$-quasi-planar graphs}.  A graph is $k$-quasi-planar if it\nadmits a planar, topological drawing in which no subset of $k+1$ edges\npairwise cross; thus a graph that is 1-quasi-planar is planar.\nVarious bounds on the number of edges in such graphs have been\ngiven~\\cite{AAPPS97,PSS94,FPS13}, culminating in:\n\\begin{theorem}[Suk and Walczak~\\cite{SW13}]\\label{thm:planar-cross}\n  A simple $n$-vertex graph admitting a topological drawing in the\n  plane in which no subset of $k+1$ edges pairwise cross has at most\n  $c_k n \\log n$ edges where $c_k$ is a constant depending only on $k$.\n\\end{theorem}\nIn fact, if one\nfollows the dependence on $k$ through Suk and Walczak's work, one finds that\n\\begin{equation}\nc_k = A^{k^k} \\text{ for a fixed constant }A \\ge 2\\label{eq:ck}\n\\end{equation}\nAs far\nas we know, such bounds have not previously been generalized to more\ngeneral surface topological drawings as we do so here.  The proof of Theorem~\\ref{thm:cross} is based on the analysis technique of Pach\net~al.~\\cite{PSS94}, but here we are able to immediately reduce the genus $g$ topological graph to a planar, topological graph, thus invoking Suk and Walczak's result~\\cite{SW13}.\n\n\\begin{theorem} \\label{thm:cross} A simple $n$-vertex graph admitting\n  a topological drawing on a surface of genus $g > 0$ in which\n  no subset of $k+1$ edges pairwise cross has at most \n  $(2g^2)^k c_k n \\log n$ when $g = O(n)$.\n\\end{theorem}\n\n\\begin{proof}\n  Let ${\\cal G}_{g,k,n}$ be the family of all graphs with at most $n$\n  vertices and admitting a genus-$g$ topological drawing in which no\n  subset of $k+1$ edges pairwise cross.  Let $m_{g,k,n}$ be the\n  maximum number of edges in any graph in ${\\cal G}_{g,k,n}$.\n\n  We aim to prove the assertion for ${\\cal G}_{g,k,n}$ that \n  \\begin{equation}\n    \\label{eq:2}\n    m_{g,k,n} \\le (2g^2)^{k}c_k n \\log n\n  \\end{equation}\n  by induction over $k$.  For $k = 1$ (and every $g$ and $n$), the\n  assertion is true since such graphs are genus-$g$ graphs and have\n  $O(n + g)$ edges which is $O(n)$ for $g = O(n)$.  For values of $n$\n  such that $n\\log n \\le (2g^2)^{k}c_k$, the assertion is\n  true since the right-hand side of Inequality~(\\ref{eq:2}) exceeds $n^2$ for all\n  such values of $n$.  We assume that $m_{g,k-1,n} \\le\n  (2g^2)^{k-1}c_{k-1} n \\log n$.\n\n  Consider a graph $G \\in {\\cal G}_{g,k,n}$ and fix a genus-$g$ topological\n  drawing of $G$ in which no subset of $k+1$ edges pairwise cross.\n  Let $\\chi$ be the number of crossings in this drawing.  We first bound $\\chi$ so we may use\n  Lemma~\\ref{lem:topo-planarize}.\n\n  Consider an edge $e$ of $G$ and let $G_e$ be the subgraph of $G$\n  consisting of all the edges crossing $e$.  Let $G_e$ inherit its\n  drawing from $G$.  Since the drawing of $G$ has no $k+1$ pairwise\n  crossing edges, the drawing of $G_e$ has no $k$ pairwise crossing\n  edges for otherwise such a set along with $e$ would witness a set of\n  $k+1$ pairwise crossing edges in the drawing of $G$.  Therefore $G_e\n  \\in {\\cal G}_{g,k-1,n}$ and so $G_e$ has at most $m_{g,k-1,n}$\n  edges.  The number of crossings on $e$ is therefore at most\n  $m_{g,k-1,n}$.  Summing over all edges of $G$, $\\chi \\le\n  \\frac{1}{2}m \\cdot m_{g,k-1,n}$ where $m$ is the number of edges in $G$.  By the inductive hypothesis,\n   \\begin{equation}\n     \\label{eq:3}\n     \\chi \\le \\frac{1}{2} m\\cdot (2g^2)^{k-1}c_{k-1} n \\log n.\n   \\end{equation}\n   \n   Let $S$ be the set of edges forming a planarizing set for $G$ guaranteed by Lemma~\\ref{lem:topo-planarize}.  By\n   Lemma~\\ref{lem:topo-planarize}, Equation~\\eqref{eq:3} and the fact\n   that $||H||_\\delta^2 \\le 2|E(H)|\\cdot|V(H)|$ for any graph $H$,\n   \\begin{equation}\n     \\label{eq:4}\n     |S| \\le \\frac{g}{2} \\sqrt{8 m\\cdot (2g^2)^{k-1}c_{k-1} n \\log n+2mn} \\le \\frac{3g}{2} \\sqrt{ m\\cdot (2g^2)^{k-1}c_{k-1} n \\log n}\n   \\end{equation}\n   where the last inequality holds for $n$ such that $2 < (2g^2)^{k-1}c_{k-1}\\log n$; these coincide with non-base-case values of $n$. \n    Let $G'$ be\n   the graph obtained by deleting $S$ from $G$.  Then $m \\le E(G') +\n   |S|$.  Since $G'$ is a $k$-quasi-planar graph on at most $n$\n   vertices, $|E(G')| \\le c_k n \\log n$ by\n   Theorem~\\ref{thm:planar-cross}.  Combining, we get\n   \\begin{equation*}\n     m \\le c_k n \\log n + \\frac{3g}{2} \\sqrt{ m\\cdot (2g^2)^{k-1}c_{k-1} n \\log n}\n   \\end{equation*}\n   Rearranging:\n   \\begin{equation} \\label{eq:f}\n     m - \\frac{3g}{2}\\sqrt{(2g^2)^{k-1}c_{k-1} n \\log n} \\sqrt{m} \\le c_k n \\log n \n   \\end{equation}\n   Let $f(m) = m - \\frac{3g}{2}\\sqrt{(2g^2)^{k-1}c_{k-1} n \\log n}\n   \\sqrt{m}$. We consider the two cases corresponding to the sign of\n   the left-hand side of (\\ref{eq:f}).\n\n   If $f(m) \\le 0$, then\n   \\[m \\le \\left(\\frac{3g}{2}\\right)^2 (2g^2)^{k-1}c_{k-1} n \\log n =\n   (2g^2)^k \\frac{9}{8}c_{k-1} n \\log n \\le (2g^2)^kc_k n \\log n,\\]\n   where the last inequality follows from ${9 \\over 8} c_{k-1} < c_k$\n   (which is clearly true given Equation~(\\ref{eq:ck})), thus proving\n   the assertion.\n   \n   We note that $f(m)$ is an increasing function for all positive\n   values of $m$ such that $f(m) > 0$.  \nWe will show that\n   \\begin{equation}\nf((2g^2)^{k}c_k n \\log n) > c_{k} n \\log n, \\label{eq:ff}\n\\end{equation}\nimplying that $m < (2g^2)^{k}c_k n \\log n$ when $f(m) > 0$, proving\nthe assertion.\n\\begin{eqnarray*}\n  f((2g^2)^{k}c_k n \\log n) &=& (2g^2)^{k}c_k n \\log n -\\frac{3g}{2}\\sqrt{(2g^2)^{k-1}c_{k-1} n \\log n} \\sqrt{(2g^2)^{k}c_k n \\log n} \\\\\n  & = & (2g^2)^{k}c_k n \\log n - \\frac{3\\sqrt{2}}{4}\\sqrt{(2g^2)^{2k}c_{k-1}c_{k}}n\\log n \\\\\n  & = & (2g^2)^{k}c_k n \\log n  \\left(1-\\frac{3\\sqrt{2}}{4}\\sqrt{\\frac{c_{k-1}}{c_k}}\\right)\\\\\n  & > & (2g^2)^{k}c_k n \\log n  \\left(1-\\frac{3\\sqrt{2}}{4}\\frac{1}{\\sqrt{2}}\\right)\\text{, since $c_k > 2 c_{k-1}$, by Equation~(\\ref{eq:ck})}\\\\\n  & = & (2g^2)^{k}c_k n \\log n  \\left(\\frac{1}{4}\\right)\\\\\n  & > & c_k n \\log n \\mbox{, for $k \\ge 2$ and $g \\ge 1$}\n \\end{eqnarray*}\nThis proves Equation~(\\ref{eq:ff}) and so the theorem.\n\\hfill \\qed \\end{proof}\n\n\n\\section{The $(p_g,q_g)$-property of genus-$g$ ball systems}\\label{sec:pq}\n\nThe proof of the fact that the ball system of a graph of genus $g$ has the $(p_g,q_g)$-property is similar to the proof of Proposition 2 in the work of Chepoi, Estellon and Vax\\`{e}s~\\cite{planarballcover}, although we have made efforts to simplify the proof here.\n\nLet $G$ be a graph of diameter at most $2R$ with an embedding on a\nsurface $\\cal S$ of genus $g$.  Let $C$ be a set of $p_g$ vertices; we\nwill define $p_g$ shortly.  Consider a set of shortest paths ${\\cal P}\n= \\{P_{ij}\\ : \\ c_i, c_j \\in C\\}$ where $P_{ij}$ is the shortest\n$c_i$-to-$c_j$ path in $G$.  We can assume, without loss of\ngenerality, that the intersection of any two of these paths is {\\em simple}, having at\nmost one component (a path or vertex), for otherwise, one path could\nbe redirected along another without compromising shortness as\nillustrated in Figure~\\ref{fig:simple}.\n\n\\begin{figure}[t]\n  \\centering\n  \\label{fig:simple}\n  \\includegraphics{simple_intersection.pdf}\n  \\caption{If $P$ and $Q$ are both shortest paths, then $P'$ must also be a shortest path between $P$'s endpoints.}\n\\end{figure}\n\n Taking the image of $P_{ij}$ on the surface for each path $P_{ij} \\in {\\cal\n P}$, we get a drawing of the complete graph $K_{p_g}$ on $\\cal S$.  We can\n make this drawing topological by a sequence of simple, local\n transformations, as illustrated in Figure~\\ref{fig:simplify}. Since we\n assumed that path intersections are simple, the first\n transformation modifies the drawing to achieve the first and third\n properties of a topological drawing and the second transformation\n modifies the drawing to achieve the second property of a topological\n drawing.  These transformations {\\em respect intersection} so far as\n that, in the final drawing of $K_{p_g}$, the images of two edges of\n $K_{p_g}$ share a point if and only if the corresponding paths share\n a vertex in $G$.\n\n\\begin{figure}[t]\n  \\centering\n  \\label{fig:simplify}\n  \\includegraphics{topological.pdf}\n  \\caption{Making a drawing topological; full details are given by Felsner~\\cite{Felsner04}.}\n\\end{figure}\n\nSince the drawing of $K_{p_g}$ is a topological drawing on surface $\\cal S$\nof genus $g$, we can use Theorem~\\ref{thm:cross} to guarantee that, for $p_g$ sufficiently large (and depending only on $g$), this drawing contains a subset of at least\n$2q_g-3$ edges that pairwise cross.  Likewise, since\nthe drawing of $K_{p_g}$ respects intersections, there must be a\nsubset ${\\cal P}'$ of at least $2q_g-3$ paths of $\\cal P$ that pairwise\nintersect.  We pick the {\\em midpoint} of a $c_i$-to-$c_j$ path\n$P_{ij} \\in {\\cal P}$ to be any vertex $m_{ij}$ that is in $B(c_i)\\cap\nB(c_j)$; since diameter of the graph is at most $2R$, the paths $\\cal\nP$ are shortest and the balls have radius $R$, such a point always\nexists.\n\n\\begin{claim}\n  For any two paths $P_{ij}, P_{k\\ell} \\in {\\cal P}'$, either $m_{ij} \\in B(c_k)\\cup B(c_\\ell)$ or $m_{k\\ell}\n\\in B(c_i) \\cup B(c_j)$.\n\\end{claim}\n\n\\begin{proof}\n  Let $x$ be a vertex shared by both $P_{ij}$ and $P_{k\\ell}$.\n  Assume, w.l.o.g., that $c_i$ is the closest of the endpoints of\n  $P_{ij}$ and $P_{k\\ell}$ ($\\{c_i,c_j,c_k,c_\\ell\\}$) to $x$.  Also\n  assume, w.l.o.g., that $x$ is in the $c_k$-to-$m_{k\\ell}$ subpath of\n  $P_{k\\ell}$.  Since $m_{k\\ell} \\in B(c_k)$ the distance from\n  $m_{k\\ell}$ to $x$ to $c_k$ is at most $R$ and since $c_i$ is closer\n  to $x$ than $c_k$, then the distance from $m_{k\\ell}$ to $x$ to\n  $c_i$ is also at most $R$, therefore $m_{k\\ell} \\in B(c_i)$.\n\\hfill \\qed \\end{proof}\n\nSince this claim holds for every pair of paths, by an averaging\nargument, there must be some path $P_{ij}$ whose midpoint is contained\nin the ball centered at the endpoint of at least $\\frac{1}{|{\\cal\n    P}'|} \\cdot {|{\\cal P}'| \\choose 2} = \\frac{1}{2}(|{\\cal P}'|-1)\n\\ge q_g-2$ paths.  Since $m_{ij}$ is additionally contained in $B(c_i)\n\\cap B(c_j)$, $m_{ij}$ is a point contained in $q_g$ balls, showing\nthat the ball system for $G$ has the $(p_g,q_g)$-property.\n\n\n\n\\section{Handling apices and toward minor-excluded graphs} \\label{sec:apex}\n\nThe Graph Minor Structure Theorem is one of many results of Robertson\nand Seymour leading to the Graph Minor theorem.  The Graph Minor\nStructure Theorem shows that for a fixed graph $H$, any graph\nexcluding $H$ as a minor is composed of graphs that, after the removal\nof a fixed number of vertices, can be embedded on a surface in which\n$H$ cannot be embedded with a fixed number of {\\em vortices}\n(described below).  These subgraphs are glued together in a tree-like\nstructure called a tree decomposition.\n\nWe are able to show that we can {\\em remove} the apices, so to speak,\nof one of these graphs (Section~\\ref{sec:rem-ap}) and that there is\none subgraph within which every other vertex is distance $R$ \n(Section~\\ref{sec:central}).  Of course, this subgraph may be quite\nlarge, but since this subgraph is nearly embeddable on some surface,\nafter removal of the apices, it may be possible to use arguments\nsimilar to those in Section~\\ref{sec:pq}.  We discuss this further in\nSection~\\ref{sec:mf-decomp}.\n\n\\subsection{Removing apices} \\label{sec:rem-ap}\n\nWe show something stronger than that of {\\em removing apices from\n  bounded genus graphs}:\n\n\\begin{lemma}\\label{lem:apex-removal}\n  Let $\\cal G$ be a class of graphs whose ball systems have\n  VC-dimension at most $q-1$ and satisfy the $(p,q)$-property.  Then\n  there is a constant $\\rho$ such that any graph in $\\cal G$ with an\n  additional $\\alpha$ apices and diameter at most $2R$ can be covered\n  by at most $\\rho+\\alpha$ balls of radius $R$.\n\\end{lemma}\n\n\\begin{proof}\n  Let $G$ be a graph such that for a subset of at most $\\alpha$ vertices $A$, $G\n  \\setminus A \\in {\\cal G}$.  Let $\\cal B$ be the ball system for $G$\n  and let ${\\cal B}'$ be the subset of those balls that do not\n  intersect $A$.  The VC-dimension of ${\\cal B}'$ is at most that of\n  ${\\cal B}$, which is at most $q-1$.  Likewise, since $\\cal B$ has\n  the $(p,q)$-property, so does ${\\cal B}'$.  By the Fractional Helly\n  Theorem, it follows that ${\\cal B}'$ has a hitting set of size at\n  most $\\rho$; this hitting set along with $A$ is a hitting set for\n  $\\cal B$. \n\\hfill \\qed \\end{proof}\n\n\n\\subsection{The central node of a tree decomposition} \\label{sec:central}\n\nA tree decomposition $\\cal T$ of a graph $G = (V,E)$ is a pair $(T,\n{\\cal X})$ where $T$ is a tree and $\\cal X$ is a family of subsets (or\n{\\em bags}) of $V$ such that:\n\\begin{itemize}\n\\item Each node $a$ of $T$ has a corresponding subset $X_a \\in {\\cal\n    X}$ and $\\cup_{X \\in {\\cal X}} X = V$;\n\\item For every edge $uv \\in E$ there is a bag $X \\in {\\cal X}$ such\n  that $u,v \\in X$.\n\\item For any three nodes $a,b,c \\in T$ such that $b$ is on the\n  $a$-to-$c$ path in $T$, $X_a \\cap X_c \\subseteq X_b$.\n\\end{itemize}\nWe refer to the {\\em nodes} of $T$ and {\\em vertices} of $G$ to avoid\nconfusion.  The width of a tree decomposition $(T,{\\cal X})$ is\n$\\max_{X \\in{\\cal X}} |X|-1$.  Tree decompositions are not unique.\nThe treewidth of a graph is the minimum possible width of a tree\ndecomposition of the graph.\n\nWe show that given a tree decomposition of a graph of diameter $2R$,\nthere is a node $a$ of the tree decomposition such that every vertex\nin the graph is within distance $R$ of some vertex in $X_a$.  This is\nsimilar to Theorem~5 by Gavoille et~al.~\\cite{GPRS01}, but we are\nspecific about the node of interest in the tree decomposition.  We\ninclude the proof below for completeness.\n\n\\begin{theorem}[Central node] \\label{thm:central-node}\n  There is a node $v$ of a tree decomposition ${\\cal T} = (T,{\\cal\n    X})$ of a graph $G$ with diameter at most $2R$ such that every vertex of\n  $G$ is within distance $R$ of some vertex in $X_v$; i.e.\\ $d(x,X_v)\n  \\le R$ for every vertex $x$ of $G$.\n\\end{theorem}\n\nConsider a node $u$ of $T$ and the corresponding bag $X_u \\in {\\cal\n  X}$. Removing $u$ from $T$ and $X_u$ from $G$ results in $k \\ge 1$\nsubgraphs, each with a tree decomposition derived from ${\\cal T}$.\nFormally, let $T^1_u, \\ldots, T^k_u$ be the components of $T \\setminus\n\\{u\\}$.  Let ${\\cal X}_u^j$ be the bags corresponding to nodes of\n$T_u^j$ with the vertices in $X_u$ removed: ${\\cal X}_u^j = \\{ X_v\n\\setminus X_u \\ : \\ v \\in T_u^j\\}$.  Let $V_u^j$ be the vertices in\nthe bags corresponding to nodes of $T_u^j$ with $X_v$ removed: $V_u^j\n= \\cup {\\cal X}_u^i$. ${\\cal T}_u^j = (T_u^j, {\\cal\n  X}_u^j)$ is a tree decomposition of the subgraph of $G$ induced by\n$V_u^j$. Since $X_u$ is a vertex separator, any $v$-to-$w$ path in $G$\nfor $v \\in V_u^i$ and $w \\in V_u^j$ ($i \\ne j$) must contain a vertex\nof $X_u$.\n\nLet $d(x,y)$ be the shortest-path distance between $x$ and $y$ in $G$.\nFor a subset of vertices $Y$, let $d(x,Y)$ be the minimum distance\nfrom $x$ to any vertex of $Y$, so $d(x,Y) = \\min_{y \\in Y}d(x,y)$. For\nany two subsets $X$ and $Y$, let $f(X,Y)$ be the furthest vertex in\n$X$ from $Y$; i.e.\\ $f(X,Y) = \\arg\\max_{x \\in X} d(x,Y)$.\n\n\\begin{lemma}\\label{lem:unique}\n  If the distance from the furthest vertex in $V^i_u$ to $X_u$ is\n  greater than $R$ for any $i$, then for every $j \\ne i$, the distance\n  from the furthest vertex in $V^j_u$ is strictly less than $R$.\n \n\\end{lemma}\n\n\\begin{proof}\n  Let $f_i = f(V^i_u,X_u)$ and let $f_j = f(V^j_u,X_u)$ for $i \\ne j$.\n\n  Let $x$ be a vertex in $X_u$ that is on a shortest path from\n  $f_i$ Note also that since\n  $f(V^i_u,X_u)$\n  and $f(V^j_u,X_u)$ are both vertices in $G$,\n  $d(f(V^i_u,X_u),f(V^j_u,X_u)) \\le 2R$. So we have:\n  \\begin{eqnarray*}\n    2R & \\ge & d(f(V^i_u,X_u),f(V^j_u,X_u)) \n    \\\\&=& d(f(V^i_u,X_u),x)+d(f(V^j_u,X_u),x)\\\\\n    &\\ge& d(f(V^i_u,X_u),X_u)+d(f(V^j_u,X_u),X_u) \\\\\n    &> &R+ d(f(V^j_u,X_u),X_u)\n\\end{eqnarray*}\n  The above then immediately implies that $d(f(V^j_u,X_u),X_u) < R$.\n\\hfill \\qed \\end{proof}\n\n  Consider the following procedure for finding the central node, starting\n  at an arbitrary node $r$:\n  \\begin{tabbing}\n  1\\qquad \\=  {\\sc search}$(r)$\\\\\n  2 \\> \\qquad\\= If $d(x,X_r) \\le R$ for all $x \\in V(G)$, return $r$. \\\\\n  3 \\> \\> Otherwise:\\\\\n  4 \\> \\> \\qquad \\= Let $p$ be a node adjacent to $r$ in $T$ such that $d(f(V_r^i,X_r),X_r) > R$ and $p \\in T_r^i$. \\\\\n  5 \\> \\> \\> {\\sc search}$(p)$.\n  \\end{tabbing}\n  \n  It is clear that if this procedure terminates, then the statement of\n  the lemma is true.  It remains to argue that the algorithm must\n  terminate.  If we we reach line 4, then, by Lemma~\\ref{lem:unique},\n  $p$ is unique.  If {\\sc search} does not terminate, then it is easy\n  to see that {\\sc search} must oscillate between two adjacent nodes\n  $p$ and $q$ of the tree decomposition: {\\sc search}$(p)$ calls {\\sc\n    search}$(q)$ and vice versa.  In this case, there must be a vertex\n  $x \\in T_q^i$ where $i$ is such that $d(f(V_q^i,X_q),X_q) > R$ and\n  $p \\in T_q^i$ and a vertex $y \\in T_p^j$ where $j$ is such that let\n  $d(f(V_p^j,X_p),X_p) > R$ and $q \\in T_p^j$.  Let $S$ be a shortest\n  $x$-to-$y$ path; by definition of $p$ and $q$, $S$ must visit a\n  vertex $a \\in X_p$ and a vertex $b \\in X_q$ (possibly $a = b$). Let\n  $m$ be a vertex closest to the middle of $S$.  Since the diameter of\n  $G$ is at most $2R$, $d(x,m)$ and $d(y,m)$ is at most $R$.\n  Therefore $b$ must come after $m$ along $S$ from $x$ to $y$ and $a$\n  must come after $m$ along $S$ from $y$ to $x$.  It must be that $m =\n  a = b$, contradicting that $d(x,X_q) > R$ and $d(y,X_p) > R$. This\n  concludes the proof of the Central Node Theorem.\n\nTheorem 5 of Gavoille et~al.'s work is an immediate corollary of Theorem~\\ref{thm:central-node}:\n\n\\begin{corollary}[Theorem 5~\\cite{GPRS01}]\n  For a graph with treewidth $tw$ and diameter $2R$, there is a set $S$ of at most\n  $tw+1$ vertices such that $d(x,S) \\leq R$ for every vertex $x$ in\n  the graph.\n\\end{corollary}\n\n\n\n\\subsection{Minor-free decompositions} \\label{sec:mf-decomp}\n\nFinally, we outline a direction for extending this result to\nminor-free graph classes and describe the challenges.\n\nRobertson and Seymour showed that for any graph $G_H$ that excludes a\nfixed minor $H$, $G_H$ has a well-defined\nstructure~\\cite{Robertson200343}.  Using the notation and terminology\nof Demaine et~al.~\\cite{HMinorFree_JACM}, the Graph Minor Structure\nTheorem states that $G_H$ is obtained by $h$-clique sums of graphs\nthat are {\\em $h$-almost embeddable} on surfaces in which $H$ cannot\nbe embedded.  A graph $G$ is\n\\emph{$h$-almost-embeddable} on a surface $S$ if:\n\\begin{itemize}\n\\item There is a set $A$ of at most $h$ vertices, called \\emph{apex}\n  vertices, such that $G\\setminus A$ can be written as a union of graphs $G_0\n  \\cup G_1 \\cup \\cdots \\cup G_h$ where $G_0$ can be cellularly\n  embedded on $S$.\n\n\\item For every $i > 0$, $G_i$ is a graph, called a {\\em vortex}, that\n  has a tree-decomposition that is a path with nodes in order\n  $x_i^1,x_i^2, \\ldots$ and width at most $h$.\n\n\\item For every $i > 0$, there is a face $F_i$ such that $u_i^1,\n  u_i^2, \\ldots$ is a subset of the boundary vertices of $F_i$ in\n  order along the boundary of $F_i$ and $u_i^j \\in X_{x_i^j}$ for all $j$.\n\\end{itemize}\nNote that since $H$ is fixed, the surfaces in which the\ncomponents of $G_H$ are almost embeddable have fixed genus.\n\nAn $h$-clique sum between graphs $A$ and $B$ identifies the vertices\nof a clique on at most $h$ vertices in $A$ and $B$ and then possibly\nremoves some edges of the clique.  The clique-sum of graphs provides a\nnatural tree decomposition.  Specifically, $G_H$ admits a tree\ndecomposition $(T,{\\cal X})$ such that for every $X \\in {\\cal X}$, the\nsubgraph of $G_H$ induced by $X$ is $h$-almost embeddable and the\nintersection of any two sets of $\\cal X$ contains at most $h$\nvertices.  Using this decomposition, we define the {\\em central subgraph}\nof $G_H$ as the subgraph of $G_H$ induced by the vertices in the\ncentral node of this tree decomposition.\n\nFocussing on this central subgraph, we can {\\em remove} the apices by\nway of Lemma~\\ref{lem:apex-removal}.  Now, in the efforts to prove the\n$(p,q)$-property for the set of balls not intersecting apices of the\ncentral subgraph, consider a set of $p$ balls for sufficiently large\n$p$.  We can assume w.l.o.g.\\ that at most one ball center is in each\nof the neighboring $H$-minor-free graphs that are clique-summed to the\ncentral subgraph; if a large number of ball centers are in one\nneighbor, then since the balls must all reach the central subgraph, a\nlarge enough number of them must share a vertex, since the clique sums\nare small.  \n\nWe can then focus on center-to-center shortest paths, as in\nSection~\\ref{sec:pq}.  For this proof technique, we need to show that\namong a set of center-to-center shortest paths, a sufficiently large\nnumber of them share an interior vertex.  While these paths must cross\nthe central subgraph and parts of them must be embedded on the surface\nthat the bulk of the central subgraph is embedded on, these paths can\nuse the clique sums and vortices to {\\em hop} over eachother, crossing\nwithout intersecting.  It does not seem possible to bound how much\nthis can happen since the {\\em number} of vortices and clique sums is\nnot bounded.  so it is likely that a more global argument, taking into\naccount the balls and not just the shortest paths between ball\ncenters, will be required in order to illustrate the $(p,q)$-property.\n\n\\section{Discussion} \\label{sec:future}\n\nThis paper presents a generalization of the ball-cover property to bounded genus graphs with a constant number of apices.  \nThis represents a significant step towards showing this result holds for all minor-free families of graphs.  This work leaves open this direct question and several others.  \n\nFor one, these results, ours and that of Chepoi et~al., do not\nevaluate the explicit number of balls required for coverage, relying\nas we do, on the Fractional Helly Theorem.  Tracing the constant\nthrough Matou\\u{s}ek's work reportedly results in a constant in excess\nof 800~\\cite{talk} while the best lower bound known is\n4~\\cite{GPRS01}.  A direct proof, bypassing the Fractional Helly\nTheorem, is likely necessary to result in more practical answers.\nLikewise, an algorithmic result is desirable, particularly if the\napplication to interval routing is to be taken seriously.\n\nFurther, since our planarizing set (Lemma~\\ref{lem:planarize}) reduces\na graph of genus $g$ to a planar graph after the removal of\n$O(g||G||_\\delta)$ edges and since Gazit and Miller give an\n$O(||G||_\\delta)$ balanced edge separator for planar graphs, we can\ncombine these results to get an $O(g||G||_\\delta)$ edge separator for\ngenus-$g$ graphs.  The obvious question is whether an $O(\\sqrt{g}||G||_\\delta)$, balanced edge separator exists for genus-$g$ graphs.  Much like Gazit and Miller's separator is a strictly tighter bound on size than the pre-existing $O(\\sqrt{\\delta_{\\max} n})$ balanced edge separator for planar graphs~\\cite{Miller86,DDSV93}, an $O(\\sqrt{g}||G||_\\delta)$, balanced edge separator for genus-$g$ graphs would be a strictly tighter bound.  Our implied  $O(g||G||_\\delta)$ edge separator results in a set of planar graphs, since the procedure starts by planarizing the graph; it is likely that a tighter bound of  $O(\\sqrt{g}||G||_\\delta)$ would not result in a set of planar graphs.\n\\iffalse\n\n\n\\subsection{TODO: edge separators}\n\nThe non-separating cycle found in each application of\nthe cut-and-contract procedure has length at most $\\frac{1}{2}||G||_f$\nby Theorem~\\ref{thm:cycle} and Lemma~\\ref{lem:cut-n-contract}.\nApplying Gazit and Miller's\nnorm-sized, balanced separator for planar\ngraphs to $G'$ gives us the following:\n\n\n\\begin{corollary}\n  There is a set of at most $s_g ||G||_f$ edges in a\n  graph $G$ of orientable genus $g$ such that cutting along these\n  edges results in a set of planar graphs, each of which having at\n  most two thirds of the weight of the faces of $G$.\n\\end{corollary}\nwhere\n\\begin{equation}\n  \\label{eq:sg}\n  s_g = g\/2 +\\sqrt{3\/4}+\\sqrt{1\/2} = \\frac{1}{2}(g+\\sqrt{3}+\\sqrt{2}).\n\\end{equation}\n\n\nOf course, applying this method to the dual $G^*$ of the graph,\nresults in an edge separator whose size is measured in terms of the\nvertex-norm of $G$:\n\n\\begin{corollary}\\label{cor:or-sep}\n  There is a set of at most $s_g ||G||_\\delta$ edges\n  in a graph $G$ of orientable genus $g$ whose removal results in a\n  set of planar graphs, each of which having at most two thirds of the weights\n  vertices of $G$.\n\\end{corollary}\n\n\\subsection{Balanced edge separators}\\label{sec:sepsep}\n\nTODO ERIN: DO YOU THINK THIS IS A LITTLE EXCESSIVE? {\\em It seems we could remove this and just mention it in one sentence, maybe in the discussion.}\n\nWe take a brief detour to generalize certain balanced, edge-separators\nfor planar graphs to graphs embedded on orientable surfaces.  Although\nthese are not needed for our main result, they are likely of\nindependent interest.\n\nBalanced separators for planar graphs are well-studied and extremely\nuseful objects, both structurally and algorithmically, most often\nproviding a means to design devide-and-conquer algorithms.  A vertex\n(resp.\\ edge) separator is any set of vertices (resp.\\ edges) whose\nremoval leaves more than one connected component.  The size of a\nseparator is the number of vertices (resp.\\ edges) in it.  A separator\nis balanced if each resulting component contains at most some constant\nfraction of the weights of the original graph's vertices, usually two-thirds;\nthroughout {\\em balanced} will refer to two-thirds.  Planar graphs\nhave $O(\\sqrt{n})$-sized, balanced vertex separators~\\cite{LT79} and\n$O(\\sqrt{\\delta n})$-sized, balanced edge\nseparators~\\cite{Miller86,DDSV93} where $n$ is the number of vertices\nin the graph and $\\delta$ is the maximum vertex degree.\n\nBecause of their utility, many generalizations of separators for\nsurface-embedded graphs exist.  For example,\nHutchinson~\\cite{h-snceg-88} proved that the shortest non-contractible\ncycle in a triangulated graph of genus $g$ has length $O(\\sqrt{n\/g} \\log g)$;\ndeleting these vertices reduces the genus by one.  Repeating until a\nplanar graph is left and then using a small, balanced planar separator\nresults in a a balanced vertex separator (where each remaining\ncomponent is planar) of size $O(\\sqrt{gn} \\log g)$.\nDjidjev~\\cite{d-st-81} and Gilbert et~al.~\\cite{ght-stgbg-84}\nindependently presented balanced vertex separators of size\n$O(\\sqrt{gn})$ for surface-embedded graphs; Eppstein~\\cite{e-dgteg-03}\nlater improved the constants and gave efficient algorithms to compute\nthe separators.  S\\'{y}kora and Vrt'o showed that genus $g$ graphs\nhave $O(\\sqrt{\\delta g n })$-sized balanced edge\nseparators~\\cite{SV93}.\n\nEdge separators depending on the maximum degree are not very useful\nexcept for low degree graphs.  Gazit and Miller gave a balanced edge\nseparator for planar graphs of size proportional to the {\\em vertex\n  norm} $||G||_\\delta$ of the graph, defined above~\\cite{GaMi90}.\nGazit and Miller showed that planar graphs admit\n$(\\sqrt{3\/4}+\\sqrt{1\/2})\\,||G||_\\delta$-sized balanced edges\nseparators~\\cite{GaMi90}.  Using the Gazit and Miller separator after\nplanarizing the graph via Lemma~\\ref{lem:planarize} gives:\n\\begin{lemma}\n  There is a set of $\\left(\\sqrt{3\\over 4}+\\sqrt{1\\over\n      2}+\\frac{g}{2}\\right)||G||_\\delta$ edges of a graph with\n  orientable genus $g$ whose removal leaves a set of planar graphs, each of which has at most 2\/3 of the weight of the vertices of the original graph.\n\\end{lemma}\n\\fi\n\n\n\\paragraph{Acknowledgements} We thank Anastasios Sidiropoulos and Mark Walsh for helpful discussions.  This material is based upon work supported by\nthe National Science Foundation under Grant Nos.\\\nCCF-0963921 and CCF-1054779.\n\n\n\\bibliographystyle{plain} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{introduction}\ncotorsion pairs (or cotorsion theory) were invented by \\cite{Sal79} in the category of abelian groups and was rediscovered by Enochs and coauthors in the 1990's. In short, a cotorsion pair in an abelian category $\\mathcal{A} $ is a pair $(\\mathcal{F} ,\\mathcal{C} )$ of classes of objects of $\\mathcal{A} $ each of which is the orthogonal complement of the other with respect to the ${\\rm{Ext}}$ functor. In recent years we have seen that the study of cotorsion pairs is especially relevant to study of covers and envelopes, particularly in the proof of the flat cover conjecture \\cite{BBE}. In 2002, Hovey established a correspondence between the theories of cotorsion pairs and model structures (Hovey's theorem \\cite{Hov02}). So the study of cotorsion pairs on the category of complexes is important, see \\cite{Gil04}, \\cite{Gil06}, \\cite{Gil08}, \\cite{EER08}, \\cite{EEI}, \\cite{EAPT}, \\cite{St}, \\cite{YD15}. Since the concept of $N$-complexes is a generalization of the ordinary complexes, it is natural to study cotorsion pairs on the category of $N$-complexes. The notion of $N$-complexes was introduced by Mayer \\cite{May42} in the his study\nof simplicial complexes and its homological theory was studied by Kapranov and\nDubois-Violette in \\cite{Kap96}, \\cite{DV98}. Besides their applications in theoretical physics \\cite{CSW07}, \\cite{Hen08}, the homological properties of $N$-complexes have become a subject of study for many authors as in, \\cite{Est07}, \\cite{Gil12}, \\cite{GH10}, \\cite{Tik02}. By an $N$-complex $\\mathbf{X}$, we mean a sequence \n$\\cdots \\rightarrow X^{n-1} \\rightarrow X^n \\rightarrow X^{n+1} \\rightarrow \\cdots $\nsuch that composition of any $N$ consecutive maps gives the zero map in $\\mathcal{A} $. \nWe can view the category of $N$-complexes as the category of representation of the quiver $A_{\\infty}^{\\infty}= \\cdots \\rightarrow v_{-1}\\rightarrow v_0\\rightarrow v_1 \\rightarrow v_2 \\rightarrow \\cdots$ with the relations that $N$ consecutive arrows compose to 0. Recently, Holm and Jorgensen in \\cite{HJ} construct model structures on the category of representations of quiver with relations, in particular for the category of $N$-complexes.\n\nIn this work we show some typical ways of getting complete cotorson pairs in the category of $N$-complexes.\nOne method for creating such pairs is by starting with two cotorsion pairs in $\\mathcal{A} $ and then\nusing these pairs to find related pairs in $\\mathbb{C} _N(\\mathcal{G} )$,  the category of $N$-complexes over a Grothendieck category $\\mathcal{G} $.  More precisely:\n\\begin{theorem}\nSuppose that $(\\mathcal{F} ,\\mathcal{C} )$ and $(\\mathcal{X} ,\\mathcal{Y} )$ are two cotorsion pairs in $\\mathcal{G} $ with $\\mathcal{F} \\subseteq\\mathcal{X} $ and the generator of $\\mathcal{G} $ is in $\\mathcal{F} $. If both $(\\mathcal{F} ,\\mathcal{C} )$ and $(\\mathcal{X} ,\\mathcal{Y} )$ are cogenerated by sets, then the induced pairs $( \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}, (\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N})^\\perp)$ and $( {}^\\perp(\\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N}), \\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N})$ are complete cotorsion pairs.\n\\end{theorem}\nFor the definition of $\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$ and $\\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N}$ see section \\ref{section 3}. This theorem recovers some results of recent work of Yang and Cao (see \\cite{YC}) and also includes the case in which the class is not closed under direct limits. As an application, we focus on particular homotopy categories and the existence of adjoint functors between them. The homotopy category $\\mathbb{K} _N(\\mathcal{A})$ of $N$-complexes of an additive category $\\mathcal{A}$ was studied by Iyama and et al. in \\cite{IKM}. In case $\\mathcal{A} ={\\rm{Mod\\mbox{-}}} R$ (the category of all left $R$-modules) they proved that $\\mathbb{K} _N^\\natural({\\rm{Prj}\\mbox{-}} R) \\cong \\mathbb{K} ^\\natural({\\rm{Prj}\\mbox{-}} \\mathbb{T} _{N-1}(R))$\nwhere $\\natural = -,b,(-,b)$ and $\\mathbb{T} _{N-1}(R)$ is the ring of triangular matrices of order $N-1$ with entries in $R$. In \\cite{BHN} the authors proved that $\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)$ is equivalent to $\\mathbb{K} ({\\rm{Prj}\\mbox{-}} \\mathbb{T} _{N-1}(R))$ whenever $R$ is a left coherent ring. This equivalence allows us to study the properties of $\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)$ from $\\mathbb{K} ({\\rm{Prj}\\mbox{-}} \\mathbb{T} _{N-1}(R))$. For instance  $\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)$ is compactly generated whenever $R$ is a left coherent ring. There is a natural question and this is whether it is possible to introduce an $N$-complex version of \\cite[Theorem 0.1]{Nee10}, \\cite[Proposition 8.1]{Nee08}. The answer is not trivial, since we do not have such an equivalence for $\\mathbb{K} _N({\\rm{Flat}\\mbox{-}} R)$ and $\\mathbb{K} ({\\rm{Flat}\\mbox{-}} \\mathbb{T} _{N-1}(R))$. Here we will show that if we consider the complete cotorsion pairs $({\\rm{Prj}\\mbox{-}} R, {\\rm{Mod\\mbox{-}}} R)$ and $({\\rm{Flat}\\mbox{-}} R, ({\\rm{Flat}\\mbox{-}} R)^\\perp)$, then we have a right adjoint functor $j^\\ast:\\mathbb{K} ({\\rm{Flat}\\mbox{-}} R)\\rightarrow \\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)$ of the natural inclusion $j_{!}:\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)\\rightarrow K_N({\\rm{Flat}\\mbox{-}} R)$, and a right adjoint functor of $j^\\ast$.\n\nThe paper is organized as follows. In section \\ref{section 2} we recall some generality on $N$-complexes and provide any background information needed through this paper such as Hill lemma. Our main result appears in section \\ref{section 3} as Theorem \\ref{theorem 3.6}. This result is generalized of \\cite[Theorem 3.13]{YC} and   \\cite[Propositions 4.8 and 4.9]{YC}. The proof of this theorem is completely different from the proof of \\cite{YC}. Finally, in section \\ref{section 4}, we will provide an $N$-complex version of the results that were shown by Neeman in the category of ordinary complexes.  \n\n\n\n\\section{preliminaries}\n\\label{section 2}\n\\subsection{The category of $N$-complexes}\nLet $\\mathcal{C}$ be an additive category. We fix a positive integer $N\\geq 2$. An $N$-complex is a diagram\n$$\\xymatrix{\\cdots \\ar[r]^{{d}^{i-1}_{\\mathbf{X}}} & X^{i} \\ar[r]^{{d}^{i}_{\\mathbf{X}}} &X^{i+1} \\ar[r]^{{d}^{i+1}_{\\mathbf{X}}}  & \\cdots }$$ \nwith $X^i \\in \\mathcal{C}$ and morphisms $d^{i}_{\\mathbf{X}} \\in {\\rm{Hom}}_{\\mathcal{C}}(X^i,X^{i+1})$ satisfying $d^N=0$. That is, composing any $N$-consecutive maps gives 0. A morphism between $N$-complexes is a commutative diagram\n$$\\xymatrix{\\cdots \\ar[r]^{{d}^{i-1}_{\\mathbf{X}}} & X^{i} \\ar[r]^{{d}^{i}_{\\mathbf{X}}} \\ar[d]^{f^i} & X^{i+1} \\ar[r]^{{d}^{i+1}_{\\mathbf{X}}} \\ar[d]^{f^{i+1}}  & \\cdots \\\\ \\cdots \\ar[r]^{{d}^{i-1}_{\\mathbf{Y}}} & Y^{i} \\ar[r]^{{d}^{i}_{\\mathbf{Y}}}& Y^{i+1} \\ar[r]^{{d}^{i+1}_{\\mathbf{Y}}} & \\cdots  }$$\nWe denote by $\\mathbb{C} _N(\\mathcal{C})$ the category of unbounded $N$-complexes. For any object $M$ of $\\mathcal{C}$ and any $j$ and $1\\leq i \\leq N$, let\n$$\\xymatrix{  D^{j}_{i}(M): \\cdots \\ar[r] & 0 \\ar[r] & X^{j-i+1} \\ar[r]^{{d}^{j-i+1}_{\\mathbf{X}}} & \\cdots \\ar[r]^{{d}^{j-2}_{\\mathbf{X}}} & X^{j-1} \\ar[r]^{{d}^{j-1}_{\\mathbf{X}}}  & X^j \\ar[r] &0 \\ar[r] & \\cdots }$$\nbe an $N$-complex satisfying $X^n=M$ and $d^{n}_{\\mathbf{X}}=1_M$ for all $(j-i+1\\leq n \\leq j)$.\n\\\\\nFor $0\\leq r<N$ and $i\\in \\mathbb{Z}$, we define\n$$ d_{\\mathbf{X},\\lbrace r\\rbrace}^{i}:=d_{\\mathbf{X}}^{i+r-1} \\cdots d_{\\mathbf{X}}^{i} $$\nIn this notation $d_{\\lbrace 1\\rbrace}^i=d^i$ and $d_{\\lbrace 0\\rbrace}^i=1_{X_i}$.\n\\begin{definition}\nLet $f:\\mathbf{X}\\longrightarrow \\mathbf{Y}$ be a morphism in $\\mathbb{C} _N(\\mathcal{C})$. Then the mapping cone $C(f)$ of $f$ define as bellow:\n\\begin{small}\n$$C(f)^m=Y^m \\oplus {\\coprod}_{i=m+1}^{m+N-1}X^i, \\qquad d^{m}_{C(f)}=\n\\left[ \\begin{array}{cccccc}\n\n\nd_{\\mathbf{Y}}^{m} & f^{m+1} & 0 & 0  &\\cdots & 0 \\\\\n0 & 0 & 1 & \\ddots & \\ddots & \\vdots \\\\\n  \\vdots & \\vdots & \\ddots & \\ddots & \\ddots & 0 \\\\\n   & 0 & \\cdots &  & 1 & 0 \\\\\n    0 & 0 & \\cdots & \\cdots & 0 & 1 \\\\\n     0 & -d_{\\mathbf{X} ,\\lbrace N-1\\rbrace}^{m+1} & -d_{\\mathbf{X} ,\\lbrace N-2\\rbrace}^{m+2} & \\cdots & \\cdots & -d_{\\mathbf{X}}^{m+N-1} \\\\\n\\end{array} \\right]  $$\n\\end{small}\n\\end{definition}\n\\begin{definition}\n\\label{def 11}\nA morphism $f:\\mathbf{X} \\longrightarrow \\mathbf{Y}$ of $N$-complexes is called null-homotopic if there exists $s^i \\in {\\rm{Hom}}_{\\mathcal{C}}(X^i,Y^{i-N+1})$ such that\n\\[ f^i=\\sum_{j=0}^{N-1} d_{\\mathbf{Y}, \\lbrace N-1-j\\rbrace}^{i-(N-1-j)}s^{i+j}d^{i}_{\\mathbf{X}, \\lbrace j\\rbrace}\\]\nWe denote by $\\mathbb{K} _N(\\mathcal{C})$ the homotopy category of unbounded $N$-complexes.\n\\end{definition}\n\\begin{definition}\nFor $\\mathbf{X}=(X^i,d^i)\\in \\mathbb{C} _N(\\mathcal{C})$, define suspension functor $\\Sigma:\\mathbb{K} _N(\\mathcal{C} )\\longrightarrow \\mathbb{K} _N(\\mathcal{C} )$ as follows:\n$$(\\Sigma \\mathbf{X})^m={\\coprod}_{i=m+1}^{m+N-1}X^i, \\qquad d^{m}_{\\Sigma \\mathbf{X}}=\n\\left[ \\begin{array}{cccccc}\n\n\n0 & 1 & 0 & 0  &\\cdots & 0 \\\\\n & 0 & \\ddots & \\ddots & \\ddots & \\vdots \\\\\n  \\vdots & \\vdots & \\ddots & \\ddots & \\ddots & 0 \\\\\n   &  &  & \\ddots & \\ddots & 0 \\\\\n    0 & 0 & \\cdots & \\cdots & 0 & 1 \\\\\n     -d_{\\lbrace N-1\\rbrace}^{m+1} & -d_{\\lbrace N-2\\rbrace}^{m+2} & \\cdots & \\cdots & \\cdot & -d^{m+N-1} \\\\\n\\end{array} \\right]     \n$$\n$$(\\Sigma^{-1} \\mathbf{X})^m={\\coprod}_{i=m-1}^{m-N+1}X^i, \\qquad d^{m}_{\\Sigma^{-1} \\mathbf{X}}=\n\\left[ \\begin{array}{cccccc}\n\n\n-d^{m-1} & 1 & 0 & \\cdots  &\\cdots & 0 \\\\\n-d_{\\lbrace 2\\rbrace}^{m-1} & 0 & 1 & \\ddots & \\ddots & \\vdots \\\\\n  \\vdots & \\vdots & \\ddots & \\ddots & \\ddots & 0 \\\\\n   &  &  & \\ddots & \\ddots & 0 \\\\\n    -d_{\\lbrace N-2\\rbrace}^{m-1} & 0 & \\cdots & \\cdots & 0 & 1 \\\\\n     -d_{\\lbrace N-1\\rbrace}^{m-1} & 0 & \\cdots & \\cdots & \\cdot & 0 \\\\\n\\end{array} \\right]     \n$$\n\\end{definition}\nLet $\\mathcal{S} _N(\\mathcal{C} )$ be the collection of short exact sequence in $\\mathbb{C} _N(\\mathcal{C} )$ of which each term is split\nexact then it is shown in \\cite{IKM} that $(\\mathbb{C} _N(\\mathcal{C} ), \\mathcal{S} _N(\\mathcal{C} ))$ is a Frobenius category and its stable\ncategory is the homotopy category $\\mathbb{K} _N(\\mathcal{C} )$ of $\\mathcal{C} $. So $\\mathbb{K} _N(\\mathcal{C} )$ together with this suspension\nfunctor is a triangulated category, see \\cite[Theorem 2.6]{IKM}.\n\nRecall that ${\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{C} )}(X,Y)$ is the group of (equivalence classes) of short exact sequences\n$0 \\rightarrow \\mathbf{Y} \\rightarrow \\mathbf{Z} \\rightarrow \\mathbf{X} \\rightarrow 0$. We let ${\\rm{Ext}}^1_{dw}(X,Y)$ be the subgroup of ${\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{C} )}(X,Y)$ consisting of those short exact sequences which are split in each degree.\n\\begin{lemma}\n\\label{lemma02}\nFor N-complex $\\mathbf{X}$ and $\\mathbf{Y}$, we have\n$${\\rm{Ext}}_{dw}^{1}(\\mathbf{Y},\\mathbf{X})\\cong {\\rm{Hom}}_{\\mathbb{K} _{N}(\\mathcal{C} )}(\\mathbf{Y},\\Sigma \\mathbf{X})  $$\n\\end{lemma}\n\\begin{proof}\nDefine a  surjective map $$\\psi:{\\rm{Hom}}_{\\mathbb{C} _N(\\mathcal{C} )}({\\Sigma}^{-1}\\mathbf{Y},\\mathbf{X})\\longrightarrow {\\rm{Ext}}_{dw}^{1}(\\mathbf{Y},\\mathbf{X})$$ by sending a morphism $f:{\\Sigma}^{-1}\\mathbf{Y}\\rightarrow \\mathbf{X}$ to the short exact sequence $0\\rightarrow \\mathbf{X} \\rightarrow C(f)\\rightarrow \\mathbf{Y}\\rightarrow 0$. Then we have the following push out diagram \n$$\\xymatrix{0 \\ar[r] & {\\Sigma}^{-1}\\mathbf{Y} \\ar[r] \\ar[d]^{f} & I({\\Sigma}^{-1}\\mathbf{Y}) \\ar[r] \\ar[d] & \\mathbf{Y} \\ar[r] \\ar[d]^{1} & 0 \\\\ 0 \\ar[r] & \\mathbf{X} \\ar[r] & C(f) \\ar[r] & \\mathbf{Y} \\ar[r]  & 0 }$$\nwhere \n$$I({\\Sigma}^{-1}\\mathbf{Y})^m= {\\coprod}_{i=m}^{m+N-1}({{\\Sigma}^{-1}\\mathbf{Y}})^i, \\qquad d^{m}_{I({\\Sigma}^{-1}\\mathbf{Y})}=\n\\left[ \\begin{array}{cccccc}\n\n\n0 & 1 & 0 & 0  &\\cdots & 0 \\\\\n0 & 0 & 1 & \\ddots & \\ddots & \\vdots \\\\\n  \\vdots & \\vdots & \\ddots & \\ddots & \\ddots & 0 \\\\\n   & 0 & \\cdots &  & 1 & 0 \\\\\n    0 & 0 & \\cdots & \\cdots & 0 & 1 \\\\\n     0 & 0 & 0 & \\cdots & \\cdots & 0 \\\\\n\\end{array} \\right]  $$\nit is easy to see that $I({\\Sigma}^{-1}\\mathbf{Y})$ is a projective-injective object in $(\\mathbb{C} _N(\\mathcal{C} ), \\mathcal{S} _N(\\mathcal{C} ))$ and all arrows are componentwise split exact sequence. But the kernel of map $\\psi$ is formed precisely by the null-homotopic morphisms $f:{\\Sigma}^{-1}\\mathbf{Y}\\rightarrow \\mathbf{X}$, since $f$ factors through $I({\\Sigma}^{-1}\\mathbf{Y})$.\n\\end{proof}\n\nLet $\\mathbf{X}$ be an $N$-complex of objects of $\\mathcal{C}$ \n$\\xymatrix{\\cdots \\ar[r]^{{d}^{i-1}_{\\mathbf{X}}} & X^{i} \\ar[r]^{{d}^{i}_{\\mathbf{X}}} &X^{i+1} \\ar[r]^{{d}^{i+1}_{\\mathbf{X}}}  & \\cdots }$, \nwe define\n$${\\rm{Z}}^{i}_{r}(\\mathbf{X}):= {\\rm{Ker}} (d^{i+r-1}_{\\mathbf{X}} \\cdots d^{i}_{\\mathbf{X}}), \\quad {\\rm{B}}^{i}_{r}(\\mathbf{X}):= {\\rm{Im}} ((d^{i-1}_{\\mathbf{X}} \\cdots d^{i-r}_{\\mathbf{X}})\n$$\n$$\n{\\rm{C}}^{i}_{r}(\\mathbf{X}):= {\\rm{Coker}}((d^{i-1}_{\\mathbf{X}} \\cdots d^{i-r}_{\\mathbf{X}}), \\quad {\\rm{H}}^{i}_{r}(\\mathbf{X}):= {\\rm{Z}}^{i}_{r}(\\mathbf{X})\/ {\\rm{B}}^{i}_{N-r}(\\mathbf{X}) \n$$\n\n\\vspace{0.1cm}\ntherefore in each degree we have $N-1$ cycle and clearly ${\\rm{Z}}^{n}_{N}(\\mathbf{X})=X^n$. \nWe also have a commutative diagram\n$$\\xymatrix{0 \\ar[r] & {\\rm{Z}}^{n}_{1}(\\mathbf{X}) \\ar[r]^{{d}^{i}_{\\mathbf{X}}} \\ar[d]^{id} & {\\rm{Z}}^{n}_{r}(\\mathbf{X}) \\ar[r]^{d} \\ar@{^{(}->}[d]^{\\qquad PB}  & {\\rm{Z}}^{n+1}_{r-1}(\\mathbf{X})\\ar@{^{(}->}[d]  \\\\ 0 \\ar[r] & {\\rm{Z}}^{n}_{1}(\\mathbf{X}) \\ar[r] & {\\rm{Z}}^{n}_{r}(\\mathbf{X}) \\ar[r]^{d} & {\\rm{Z}}^{n+1}_{r}(\\mathbf{X})  }$$ \nwhere the right square is pull-back $(2 \\leq r \\leq N-1) $.\n\n\\begin{definition}\nLet $\\mathbf{X} \\in \\mathbb{K} _N(\\mathcal{C})$. We say $\\mathbf{X}$ is $N$-exact if ${\\rm{H}}^{i}_{r}(\\mathbf{X})=0$\nfor each $i \\in \\mathbb{Z}$ and all $r=1,2,...,N-1$. We denote the full subcategory of $\\mathbb{K} _N(\\mathcal{C})$ consisting of $N$-exact complexes by $\\mathcal{E}_N(\\mathcal{C})$ .\n\\end{definition}\n\nThe following result are useful. See \\cite[Lemma 3.9]{IKM}\n\\begin{lemma}\n\\label{lemma 001}\nLet $\\mathbf{X}$ be an N-complex of objects of an abelian category $\\mathcal{C}$. For a commutative diagram \n$$\\xymatrix{ {\\rm{Z}}^{n}_{r}(\\mathbf{X}) \\ar[r]^{d^{n}_{r}} \\ar@{^{(}->}[d]^{\\iota_{r}^{n}\\qquad PB}  & {\\rm{Z}}^{n+1}_{r-1}(\\mathbf{X})\\ar@{^{(}->}[d]^{\\iota_{r-1}^{n+1}}  \\\\  {\\rm{Z}}^{n}_{r+1}(\\mathbf{X}) \\ar[r]^{d^{n}_{r+1}} & {\\rm{Z}}^{n+1}_{r}(\\mathbf{X})  }$$\nthe following hold.\n\\begin{itemize}\n\\item[(1)] $\\mathbf{X} \\in \\mathcal{E}_N(\\mathcal{C})$ if and only if $d^{n}_{r}$ is an epimorphism for any n and r.\n\\item[(2)] $\\mathbf{X}$ is homotopic to 0 if and only if $d^{n}_{r}$ is a split epimorphisms and $\\iota_{r}^{n}$ is a split monomorphisms for any n and r. \n\\end{itemize}\n\\end{lemma}\n\n\\begin{remark}\n\\label{remark 001}\nAn $N$-complex $\\mathbf{X}$ is $N$-exact if and only if there exists some $r$ with $1\\leq r\\leq N-1$ such that ${\\rm{H}}^{i}_{r}(\\mathbf{X})=0$ for each integer $i$, see \\cite{Kap96}.\n\\end{remark}\n\n\n\\begin{remark}\n\\label{remark 002}\nBy use of lemma \\ref{lemma 001}, it is easy to show that whenever $\\mathbf{X}$ is an $N$-exact complex then $\\Sigma \\mathbf{X} $ and $\\Sigma^{-1} \\mathbf{X}$ are $N$-exact complexes.\n\\end{remark}\n\\begin{lemma}\n\\label{lemma01}\nLet $\\mathcal{A} $ be an abelian category. For an object $M \\in \\mathcal{A} $, $1\\leq r \\leq N-1$ and $\\mathbf{X},\\mathbf{Y} \\in \\mathbb{C} _N(\\mathcal{A} )$ we have the following isomorphism:\n\\begin{itemize}\n\\item[(1)]${\\rm{Ext}}_{\\mathcal{A} }^{1}(M,Y^n)\\cong {\\rm{Ext}}_{\\mathbb{C} _N(\\mathcal{A} )}^1(D_{N}^{n+N-1}(M),\\mathbf{Y})$\n\\item[(2)]${\\rm{Ext}}_{\\mathcal{A} }^{1}(X^n,M)\\cong {\\rm{Ext}}_{\\mathbb{C} _N(\\mathcal{A} )}^1(\\mathbf{X},D_{N}^{n}(M))$\n\\item[(3)]${\\rm{Ext}}_{\\mathbb{C} _N(\\mathcal{A} )}^{1}(D_{r}^{n+r-1}(M),\\mathbf{Y})\\cong {\\rm{Ext}}_{\\mathcal{A} }^{1}(M,{\\rm{Z}}^{n}_{r}(\\mathbf{Y}))$ whenever $\\mathbf{Y}$ is an N-exact complex.\n\\item[(4)]${\\rm{Ext}}_{\\mathbb{C} _N(\\mathcal{A} )}^{1}(\\mathbf{X},D_{r}^{n}(M))\\cong {\\rm{Ext}}_{\\mathcal{A} }^{1}({\\rm{C}}^{r}_{n}(\\mathbf{X}),M)$ whenever $\\mathbf{X}$ is an N-exact complex.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\nSee \\cite[section 4]{GH10} or \\cite[Lemma 2.2]{YC} for more details. \n\n\\end{proof}\n\n\n\\begin{definition}\nA pair of classes $(\\mathcal{F} ,\\mathcal{C} )$ in abelian category $\\mathcal{A} $ is a cotorsion pair if the following conditions hold:\n\\begin{itemize}\n\\item[1.]${\\rm{Ext}}^1_\\mathcal{A} (F,C)=0$ for all $F\\in \\mathcal{F} $ and $C\\in \\mathcal{C} $\n\\item[2.]If ${\\rm{Ext}}^1_\\mathcal{A} (F,X)=0$ for all $F\\in \\mathcal{F} $, then $X\\in \\mathcal{C} $.\n\\item[3.]If ${\\rm{Ext}}^1_\\mathcal{A} (Y,C)=0$ for all $C\\in \\mathcal{C} $, then $Y\\in \\mathcal{F} $.\n\\end{itemize}\n\\end{definition}\nWe think of a cotorsion pair $(\\mathcal{F} ,\\mathcal{C} )$ as being ``orthogonal with respect to ${\\rm{Ext}}^1_\\mathcal{A} $. This is often\nexpressed with the notation $\\mathcal{F}^\\perp=\\mathcal{C}$ and $\\mathcal{F} = {}^\\perp\\mathcal{C}$. A cotorsion pair $(\\mathcal{F} ,\\mathcal{C} )$ is called complete\nif for every $A \\in \\mathcal{A} $ there exist exact sequences\n\\[0 \\rightarrow Y \\rightarrow W \\rightarrow A \\rightarrow 0 \\ \\ \\ {\\rm and} \\ \\ \\ 0 \\rightarrow A  \\rightarrow Y' \\rightarrow W' \\rightarrow 0,\\]\nwhere $W, W'\\in \\mathcal{F}$ and $Y, Y' \\in  \\mathcal{C}$.\nWe note that if $\\mathcal{S}$ is any class of objects of $\\mathcal{A}$ and if $\\mathcal{S}^\\perp=\\mathcal{B}$ and $\\mathcal{A}={}^\\perp\\mathcal{B}$, then $(\\mathcal{A}, \\mathcal{B})$ is a cotorsion pair. We say it is the cotorsion pair cogenerated by $\\mathcal{S}$. If there is a set $\\mathcal{S}$ that cogenerates $(\\mathcal{A}, \\mathcal{B})$, then we say that $(\\mathcal{A}, \\mathcal{B})$ is cogenerated by a set.\n\n\n\n\n\n\\subsection{Hill lemma:}\n\\label{subsect 01}\nLet $\\mathcal{G} $ be a Grothendieck category endowed with a faithful functor $U:\\mathcal{G} \\rightarrow\\textbf{Set}$, where \\textbf{Set} denotes the category of sets. By abuse of notation we write $x\\in\\mathcal{G} $ instead of\n$x \\in U(G)$, for any object $G$ in $\\mathcal{G} $. Analogously $|G|$ will denote the cardinality of $U(G)$. We\nwill also assume that there exists an infinite regular cardinal $\\lambda$ such that for each $G\\in \\mathcal{G} $ and\nany set $S\\subseteq G$ with $|S|<\\lambda$, there is a subobject $X\\subseteq G$ such that $S \\subseteq X \\subseteq G$ and $|X|<\\lambda$.\n\nGiven an infinite regular cardinal $\\kappa$. Recall that an object $X \\in \\mathcal{G} $ is called $\\kappa$-presentable if the functor ${\\rm{Hom}}_\\mathcal{G} (X,-):\\mathcal{G}  \\rightarrow \\textbf{Ab}$\npreserves $\\kappa$-filtered colimits. An object $X\\in \\mathcal{G} $ is called $\\kappa$-generated whenever ${\\rm{Hom}}_\\mathcal{G} (X,-)$ preserves $\\kappa$-filtered colimits of monomorphisms. By our assumption it is easy to see that \n$$ |X|<\\lambda \\,\\,\\,\\,\\, \\Leftrightarrow \\,\\,\\,\\,\\, X\\,\\, \\text{is}\\,\\,  \\lambda\\text{-presentable}\\,\\,\\,\\,\\, \\Leftrightarrow \\,\\,\\,\\,\\, X\\,\\, \\text{is}\\,\\,  \\lambda\\text{-generated } $$\n\n\\begin{definition}\nLet $\\mathcal{S} $ be a class of objects of $\\mathcal{G} $. An object $X\\in\\mathcal{G} $ is called $\\mathcal{S} $-filtered if\nthere exists a well-ordered direct system $(X_\\alpha, i_\\alpha\\beta|\\alpha<\\beta\\leq \\sigma)$ indexed by an ordinal number $\\sigma$ such that\n\\begin{itemize}\n\\item[(a)]$X_0=0$ and $X_\\sigma=X$,\n\\item[(b)]For each limit ordinal $\\mu\\leq \\sigma$, the colimit of system $(X_\\alpha, i_{\\alpha\\beta}|\\alpha<\\beta\\leq \\mu)$ is precisely\n$X_\\mu$, the colimit morphisms being $i_{\\alpha\\mu}:X_\\alpha \\rightarrow X_\\mu$,\n\\item[(c)]$i_{\\alpha\\beta}$ is a monomorphism in $\\mathcal{G} $ for each $\\alpha < \\beta \\leq \\sigma$,\n\\item[(d)]${\\rm{Coker}} i_{\\alpha\\alpha+1}\\in \\mathcal{S} $ for each $\\alpha<\\sigma$\n\\end{itemize}\n\\end{definition}\nThe direct system $(X_\\alpha, i_\\alpha\\beta)$ is then called an $\\mathcal{S} $-filtration of $X$. The class of all $\\mathcal{S} $-filtered objects in $\\mathcal{G} $ is denoted by Filt-$\\mathcal{S} $.\n\nThe Hill lemma is a way of creating a plentiful supply of a module with a given filtration, but where these submodules have nice properties. This result, whose idea is due to Hill \\cite{Hill} and version of which appeared in \\cite{FL}. In the following we state the Hill lemma for Grothendieck category which is known as the generalized Hill lemma, see \\cite[Theorem 2.1]{St}.\n\\begin{theorem}\n\\label{theorem Hill}\nLet $\\mathcal{G} $ be as above and $\\kappa$ be a regular infinite cardinal such that $\\kappa\\geq \\lambda$. Suppose that $\\mathcal{S}$ is a set of $\\kappa$-presentable objects and $X$ is an object possessing an $\\mathcal{S} $-filtration $(X_\\alpha\\,\\,|\\,\\, \\alpha\\leq \\sigma)$ for some ordinal $\\sigma$. Then there is a complete sublattice $\\mathcal{L}$ of $(\\mathcal{P}(\\sigma),\\cup,\\cap)$ and $\\ell: \\mathcal{L}\\rightarrow \\text{Subobj}(X)$ which assigns to each $S\\in \\mathcal{L}$ a subobject $\\ell(S)$ of $X$, such that the following hold:\n\\begin{itemize}\n\\item[(H1)] For each $\\alpha\\leq\\sigma$ we have $\\alpha=\\lbrace \\gamma \\, \\, | \\, \\, \\gamma<\\alpha\\rbrace\\in \\mathcal{L}$ and $\\ell(\\alpha)=X_\\alpha$.\n\\item[(H2)] If $(S_i)_{i\\in I}$ is a family of elements of $\\mathcal{L}$, then $\\ell(\\cup S_i)=\\sum \\ell(S_i)$ and $\\ell(\\cap S_i)=\\cap \\ell(S_i)$.\n\\item[(H3)] If $S,T\\in \\mathcal{L}$ are such that $S\\subseteq T$, then the object $N=\\ell(T)\/\\ell(S)\\in \\text{Filt-}\\mathcal{S} $.\n\\item[(H4)] For each $\\kappa$-presentable subobject $Y\\subseteq X$, there is $S\\in \\mathcal{L}$ of cardinal $< \\kappa$( so $\\ell(S)$ is $\\kappa$-presentable by (H3)) such that $Y\\subseteq\\ell(S)\\subseteq X$.\n\\end{itemize}\n\\end{theorem}\nLet $\\mathcal{H}=\\lbrace \\ell(S)\\, | \\, S\\in \\mathcal{L}\\rbrace$. We call $\\mathcal{H}$ as the Hill class of subobjects of $X$ relative to $\\kappa$.\n\n\\begin{corollary}\nIf $N\\in \\mathcal{H}$ and $M$ is a $\\kappa$-presentable subobject of $X$, then there exists $P\\in \\mathcal{H}$ such that $N+M\\subseteq P$ and $P\/N$ is $\\kappa$-presentable.\n\\end{corollary}\n\\begin{proof}\nBy using theorem \\ref{theorem Hill} (H4), we can find $S\\in \\mathcal{L}$ of cardinal $< \\kappa$ such that $M\\subseteq \\ell(S)$. Denoting $W=\\ell(S)$, $P=N+W$ and combining (H2) and (H3) of theorem \\ref{theorem Hill} with \\cite[corollary A.5]{St} we observe that $P\\in \\mathcal{H}$ and $P\/N$ is $\\kappa$-presentable.\n\\end{proof}\n\nWe need the following lemma and theorem.\n\\begin{lemma}\n\\label{lemma2.14}\nLet $\\kappa$ be a regular infinite cardinal such that $\\kappa>\\lambda$. Let $\\mathbf{X}\\subseteq\\mathbf{Y}$ be $N$-exact complexes. For each $i\\in \\mathbb{Z} $, let $M^i$ be a $\\kappa$-presentable object of $Y^i$. Then there exists an $N$-exact complex $\\mathbf{E}$ such that $\\mathbf{X} \\subseteq \\mathbf{E}\\subseteq \\mathbf{Y}$ and for each $i\\in \\mathbb{Z} $, $M^i+X^i\\subseteq E^i$ and the object $E^i\/X^i$ is $\\kappa$-presentable.\n\\end{lemma}\n\\begin{proof}\nWe use the zig-zag technique to construct $\\mathbf{E}$. First, consider the particular case $\\mathbf{X}=0$. We will construct $\\mathbf{E}$ as the union of an increasing sequence of $N$-subcomplexes\n$$\\mathbf{C}_0\\subseteq \\mathbf{C}_1 \\subseteq \\mathbf{C}_2 \\subseteq \\cdots  $$\nof $\\mathbf{E}$ where $M^i\\subseteq C_0^i$, $|C_n^i| \\leq \\kappa$ and ${\\rm{Z}}_r^i(\\mathbf{C}_n)\\subseteq {\\rm{B}}_{N-r}^i(\\mathbf{C}_{n+1})$ for all $1\\leq r \\leq N-1$ and $i,n\\in \\mathbb{Z} $. Then if $\\mathbf{E}=\\cup_{n\\in \\mathbb{Z} }\\mathbf{C}_n$, we have ${\\rm{Z}}_r^i(\\mathbf{E})=\\cup_{n\\in \\mathbb{Z} }{\\rm{Z}}_r^i(\\mathbf{C}_n)\\subseteq \\cup_{n\\in \\mathbb{Z} }{\\rm{B}}_{N-r}^i(\\mathbf{C}_n)\\subseteq {\\rm{B}}_{N-r}^i(\\mathbf{E})$. So ${\\rm{Z}}_r^i(\\mathbf{E})={\\rm{B}}_{N-r}^i(\\mathbf{E})$ for all $1 \\leq r \\leq N-1$ and $i\\in \\mathbb{Z} $, hence $\\mathbf{E}\\in \\mathcal{E}_N(\\mathcal{G} )$. In this case clearly $M^i\\subseteq E^i$ and $|\\mathbf{E}| \\leq \\kappa$, since $|\\mathbf{C}_n| \\leq \\kappa$. Let $\\mathbf{C}_0=(C_0^i)$ be such that $C_0^{i}=M^i+\\sum_{k=1}^{N-1} d^{i-N+k}_{\\lbrace  N-k \\rbrace}(M^{i-N+k})$. Then $\\mathbf{C}_0$ is a subcomplex of $\\mathbf{Y}$ and clearly $M^i\\subseteq C_0^i$ and $|\\mathbf{C}_0| \\leq \\kappa$. Having constructed $\\mathbf{C}_n$ with $|\\mathbf{C}_n|\\leq \\kappa$, we want to construct $\\mathbf{C}_{n+1}$ with $\\mathbf{C}_n\\subseteq \\mathbf{C}_{n+1}$ such that ${\\rm{Z}}_r^i(\\mathbf{C}_n)\\subseteq{\\rm{B}}_{N-r}^i(\\mathbf{C}_{n+1})$ and $|\\mathbf{C}_{n+1}|\\leq \\kappa$. \n\nFor each $i\\in \\mathbb{Z} $, we have ${\\rm{Z}}_r^i(\\mathbf{C}_n)\\subseteq {\\rm{Z}}_r^i(\\mathbf{Y})={\\rm{B}}_{N-r}^i(\\mathbf{Y})$. So  by our assumption on $\\kappa$ we can find a subobject $S^{i-N+r}\\subseteq Y^{i-N+r}$ such that ${\\rm{Z}}_r^i(\\mathbf{C}_n)\\subseteq d^{i-N+r}_{\\lbrace N-r \\rbrace}(S^{i-N+r})$ for all $1\\leq r \\leq N-1$. Now define $C_{n+1}^i= C_n^i+S^i+\\sum_{k=1}^{N-1} d^{i-N+k}_{\\lbrace  N-k \\rbrace}(S^{i-N+k})$ for all $i\\in \\mathbb{Z} $. Clearly $\\mathbf{C}_n\\subseteq \\mathbf{C}_{n+1}$ and by construction ${\\rm{Z}}_r^i(\\mathbf{C}_n)\\subseteq {\\rm{B}}_{N-r}^i(\\mathbf{C}_{n+1})$ so finally we have the  desire $\\mathbf{E}\\subseteq\\mathbf{Y}$.\nIn case $\\mathbf{X}\\neq 0$ let $\\overline{\\mathbf{Y}}=\\mathbf{Y}\/\\mathbf{X} $ and $\\overline{M^i}=(M^i+X^i)\/X^i$. According to the previous part, there is an $N$-exact complex $\\overline{\\mathbf{E}}\\subseteq \\overline{\\mathbf{Y}}$ and for each $i\\in \\mathbb{Z} $, $\\overline{M^i}\\subseteq \\overline{E^i}$, and the object $\\overline{E^i}$ is $\\kappa$-presentable. Then $\\overline{\\mathbf{E}}=\\mathbf{E}\/\\mathbf{X}$ for an $N$-exact subcomplex $\\mathbf{X}\\subseteq\\mathbf{E}\\subseteq\\mathbf{Y}$, and $\\mathbf{E}$ clearly has the required properties. \n\\end{proof}\n\n\n\\begin{theorem}\n\\label{theorem 2.12}\nLet $\\kappa$ be an uncountable regular cardinal such that $\\kappa > \\lambda$. Let $(\\mathcal{F} , \\mathcal{C} )$ be a\ncotorsion pair in $\\mathcal{G} $ such that $\\mathcal{F} $ contains a family of $\\lambda$-presentable generators of $\\mathcal{G} $. Then the following conditions are equivalent:\n\\begin{itemize}\n\\item[(1)]The cotorsion pair $(\\mathcal{F} , \\mathcal{C} )$ is cogenerated by a class of $\\kappa$-presentable objects in $\\mathcal{G} $\n\\item[(2)]Every object in $\\mathcal{F} $ is $\\mathcal{F} ^\\kappa$-filtered, where $\\mathcal{F} ^\\kappa$ is the class of all $\\kappa$-presentable objects in $\\mathcal{F} $.\n\\end{itemize}\n\\end{theorem}\n\\begin{proof}\nWe refer to \\cite[ Theorem 2.1]{EEI}.\n\\end{proof}\n\n\\section{Induced cotorsion pairs in $\\mathbb{C} _N(\\mathcal{A} )$}\n\\label{section 3}\nIn this section we show some typical ways of getting complete cotorson pairs in $\\mathbb{C} _N(\\mathcal{A} )$. One method for creating such pairs is by starting with two cotorsion pairs in $\\mathcal{A} $ and then using these pairs to find related pairs in $\\mathbb{C} _N(\\mathcal{A} )$. We start with the following proposition:\n\\begin{proposition}\n\\label{prop31}\nLet $\\mathcal{A} $ be an abelian category with injective cogenerator $J$ and $\\mathbf{X}$ be an $N$-complex. If every chain map $\\mathbf{X}\\rightarrow D^{i-r+1}_r(J)$ extends to $D^{i+N-r}_N(J)$ for each $i\\in \\mathbb{Z} $ and $1\\leq r\\leq N-1$ then $\\mathbf{X}$ is an $N$-exact complex.\n\\end{proposition}\n\\begin{proof}\nBy remark \\ref{remark 001}, we show that ${\\rm{H}}^i_1(\\mathbf{X})={\\rm{Z}}^i_1(\\mathbf{X})\/{\\rm{B}}^i_{N-1}(\\mathbf{X})$ is zero. Consider the monomorphism map $\\imath: X^i\/{\\rm{B}}^i_1(\\mathbf{X})\\hookrightarrow J$. Since ${\\rm{B}}^i_{N-1}(\\mathbf{X})\\subseteq {\\rm{B}}^i_1(\\mathbf{X})$, so we set $t$ as the following composition\n$$\\xymatrix{t: X^i\/{\\rm{B}}^i_{N-1}(\\mathbf{X}) \\ar@{^{(}->}[r]^{q} & X^i\/{\\rm{B}}^i_1(\\mathbf{X}) \\ar@{^{(}->}[r]^{\\,\\,\\quad\\imath}& J }$$\nNow consider $f:\\mathbf{X}\\rightarrow D^i_1(J)$ with $f^n=0$ for $n\\neq i$ and $f^i$ is the composition of morphism $\\pi^i:X^i\\rightarrow X^i\/{\\rm{B}}^i_{N-1}(\\mathbf{X})$ and $t$. It is easy to check that $f$ is a morphism of $N$-complexes. By assumption we can extend this morphism to a morphism $g:\\mathbf{X}\\rightarrow D^{i+N-1}_N(J)$, i.e. we have the following commutative diagram\n$$\\xymatrix{ & \\mathbf{X} \\ar[d]^{f} \\ar[dl]_g & \\\\\n            D^{i+N-1}_N(J) \\ar[r]^h & D^i_1(J) \\ar[r] & 0  }$$\nPut $d^i=q^i \\pi^i$. Then we have $t\\pi^i=f^i=h^i g^i=g^i=g^{i-1}d^i=g^{i-1}q^i\\pi^i$. Hence $t=g^{i-1}q^i$, since $\\pi^i$ is epimorphism. This implies that $q^i$ is monomorphism, therefore ${\\rm{Z}}^i_1(\\mathbf{X})=\\ker d^i=\\ker(q^i\\pi^i)=\\ker(\\pi^i)={\\rm{B}}^i_{N-1}(\\mathbf{X})$. Hence ${\\rm{H}}^i_1(\\mathbf{X})=0$.\n\\end{proof}\n\n\\begin{definition}\t\nLet $\\mathcal{A} $ be an abelian category. Given two classes of objects $\\mathcal{X} $ and $\\mathcal{F} $ in\n$\\mathcal{A} $ with $\\mathcal{F} \\subseteq \\mathcal{X} $. We denote by $\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$ the class of all $N$-exact complexes $\\mathbf{F}$ with each degree\n$F^i\\in \\mathcal{F} $ and each cycle ${\\rm{Z}}_{r}^{i}(\\mathbf{X}) \\in \\mathcal{X} $ for all $1\\leq r\\leq N-1$ and $i\\in \\mathbb{Z} $.\n\\end{definition}\n\\begin{proposition}\n\\label{prop 3.3}\nLet $\\mathcal{A} $ be an abelian category with injective cogenerator $J$. Let $(\\mathcal{F} ,\\mathcal{C} )$ and $(\\mathcal{X} ,\\mathcal{Y} )$ be two cotorsion pairs with $\\mathcal{F} \\subseteq \\mathcal{X} $ in $\\mathcal{A} $. Then\n$( \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}, (\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N})^\\perp)$ is a cotorsion pair in $\\mathbb{C} _N(\\mathcal{A} )$ and $(\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N})^\\perp$ is the class of all $N$-complexes $\\mathbf{C}$\nfor which each $C^i \\in \\mathcal{C} $ and for each map $\\mathbf{F}\\rightarrow \\mathbf{C}$ is null-homotopic whenever $\\mathbf{F}\\in\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$.\n\\end{proposition}\n\\begin{proof}\nLet $\\mathcal{W} $ be the class of all $N$-complexes $\\mathbf{C}$ for which each $C^i\\in \\mathcal{C} $ and for which each map $\\mathbf{F}\\rightarrow \\mathbf{C}$ is null-homotopic whenever $\\mathbf{F}\\in \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$. It is easy to check that $\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$ is closed under $\\Sigma$ and $\\Sigma^{-1}$. Hence, by \\cite[Corollary 2.16]{YD} we can say that $\\mathcal{W} $ is closed under taking suspensions. Now suppose that $C\\in \\mathcal{C} $ and $\\mathbf{F}\\in \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$. By lemma \\ref{lemma01} (2) we have ${\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{A} )}(\\mathbf{F}, D^{i}_{N}(C))\\cong {\\rm{Ext}}^1_\\mathcal{A}(F^{i},C)=0$. But ${\\rm{Ext}}^1_{dw}(\\mathbf{F},D^i_{N}(C))={\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{A} )}(\\mathbf{F}, D^{i}_{N}(C))$, so by lemma \\ref{lemma02} we can say that $D^{i}_{N}(C)$ belongs to $\\mathcal{W} $ for each $i\\in \\mathbb{Z} $.\n Similarly, for any $\\mathbf{F}\\in \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$, by lemma \\ref{lemma01}(4) we get that ${\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{A} )}(\\mathbf{F}, D^{i}_{r}(C))\\cong {\\rm{Ext}}^1_\\mathcal{A}({\\rm{C}}^{i}_{r}(\\mathbf{F}),C)=0$, since ${\\rm{C}}^{i}_{r}(\\mathbf{F})=F^i\/{\\rm{B}}^{i}_{r}(\\mathbf{X})\\cong {\\rm{Z}}^{i+1}_r(\\mathbf{F})\\in \\mathcal{X} $. \n\nNow we show that $(\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N},\\mathcal{W} )$ is a cotorsion pair. First of all suppose that $\\mathbf{F}\\in \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$ and $\\mathbf{W}\\in \\mathcal{W} $. By assumption any $\\zeta: 0 \\rightarrow \\mathbf{W} \\rightarrow \\mathbf{A} \\rightarrow \\mathbf{F} \\rightarrow 0$ as an object of ${\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{A} )}(\\mathbf{F}, \\mathbf{W})$ is degreewise split, so belongs to ${\\rm{Ext}}^1_{dw}(\\mathbf{F},\\mathbf{W})$. But by lemma \\ref{lemma02} ${\\rm{Ext}}^1_{dw}(\\mathbf{F},\\mathbf{W})=0$. Hence ${\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{A} )}(\\mathbf{F}, \\mathbf{W})=0$. Next assume that ${\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{A} )}(\\mathbf{F}, \\mathbf{A})=0$ for all $\\mathbf{F}\\in \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$. We will show that $\\mathbf{A}\\in \\mathcal{W} $. To this point let $Z\\in \\mathcal{F} $. By lemma \\ref{lemma01}(1) ${\\rm{Ext}}^1_\\mathcal{A}(Z,A^i)\\cong{\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{A} )}(D^{i+N-1}_N(Z),\\mathbf{A} )=0$, Since $D^{i+N-1}_N(Z)$ is clearly belongs to $\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$. Thus $A^i\\in \\mathcal{C} $. Now let $u:\\mathbf{F}\\rightarrow \\mathbf{A}$ be a morphism in $\\mathbb{C} _N(\\mathcal{A} )$ where $\\mathbf{F}\\in \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$. Clearly we have that ${\\rm{Ext}}^1_{dw}(\\mathbf{F},\\Sigma^{-1}\\mathbf{A})={\\rm{Ext}}^1_{dw}(\\Sigma\\mathbf{F},\\mathbf{A})$ and the last group equals to 0 since $\\Sigma\\mathbf{F}\\in \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$ so by lemma \\ref{lemma02} we can say that $u$ is null-homotopic and hence $\\mathbf{A}\\in \\mathcal{W} $. Finally, assume that ${\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{A} )}(\\mathbf{A}, \\mathbf{W})=0$ for all $\\mathbf{W}\\in \\mathcal{W} $. We will show that $\\mathbf{A}\\in \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$. Let $C\\in \\mathcal{C} $. As we know before $D^i_N(C)\\in \\mathcal{W} $, hence we have ${\\rm{Ext}}^1_\\mathcal{A}(A^i,C)\\cong{\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{A} )}(\\mathbf{A},D^i_N(C) )=0$ and so $A^i\\in \\mathcal{F} $. It is easy to check that $\\mathcal{F}  \\subseteq \\mathcal{X} $ if and only if $\\mathcal{Y} \\subseteq \\mathcal{C} $. Also we know that if $Y\\in \\mathcal{Y} $ then $D^i_r(Y)\\in \\mathcal{W} $. So ${\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{A} )}(\\mathbf{A}, D^i_r(Y))=0$. Consider  the exact sequence $0\\rightarrow D^{i+N-r}_{N-r}(J)\\rightarrow D^{i+N-r}_N(J) \\rightarrow D^{i-r+1}_r(J) \\rightarrow 0$. We apply the convariant functor ${\\rm{Hom}}_{\\mathbb{C} _N(\\mathcal{A} )}(\\mathbf{A},-)$ to the sequence, so we have the following exact sequence\n$$ {\\rm{Hom}}_{\\mathbb{C} _N(\\mathcal{A} )}(\\mathbf{A},D^{i+N-r}_N(J)) \\longrightarrow {\\rm{Hom}}_{\\mathbb{C} _N(\\mathcal{A} )}(\\mathbf{A},D^{i-r+1}_r(J)) \\longrightarrow 0$$   \nHence, by proposition \\ref{prop31} we can say that $\\mathbf{A}$ is an $N$-exact complex. \n\nOn the other hand ${\\rm{Ext}}^1_\\mathcal{A}({\\rm{C}}^{i}_{r}(\\mathbf{A}),Y)\\cong{\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{A} )}(\\mathbf{A}, D^{i}_{r}(Y))=0$. Hence ${\\rm{C}}^{i}_{r}(\\mathbf{A})\\in \\mathcal{X} $ and therefore ${\\rm{Z}}^{i+1}_r(\\mathbf{A})\\in \\mathcal{X} $, since ${\\rm{C}}^{i}_{r}(\\mathbf{A})\\cong{\\rm{Z}}^{i+1}_r(\\mathbf{A})$. So $\\mathbf{A}\\in \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$ and we are done.\n\\end{proof}\nWe also have the following result.\n\\begin{proposition}\n Let $\\mathcal{A} $ be an abelian category with generator $G$ and $(\\mathcal{F} ,\\mathcal{C} )$ and $(\\mathcal{X} ,\\mathcal{Y} )$ be two cotorsion pairs with $\\mathcal{F} \\subseteq \\mathcal{X} $ in $\\mathcal{A} $. Then\n$( {}^\\perp(\\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N}), \\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N})$ is a cotorsion pair in $\\mathbb{C} _N(\\mathcal{A} )$ and ${}^\\perp(\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N})$ is the class of all $N$-complexes $\\mathbf{X}$\nfor which each $X^i \\in \\mathcal{X} $ and for each map $\\mathbf{X}\\rightarrow \\mathbf{Y}$ is null-homotopic whenever $\\mathbf{Y}\\in\\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N}$.\n\\end{proposition} \n\\begin{proof}\nIt is dual to the proof of Proposition \\ref{prop 3.3}.\n\\end{proof}\nIn the papers \\cite{Gil04,Gil08} Gillespie introduced some classes of complexes and find new cotorsion pairs in the category of complexes. In similar manner we can define these classes in the category of $N$-complexes. In the following, we summarize these several classes of $N$-complexes.\n\\begin{definition}\n\\label{def 35}\nLet $(\\mathcal{F},\\mathcal{C})$ be a cotorsion pair in $\\mathcal{A} $. Let $\\mathcal{E}_N$ be a class of $N$-exact complexes. We will consider the following subclasses of $\\mathbb{C} _N(\\mathcal{A} )$:\n\\begin{itemize}\n\\item[(1)]The class of $\\mathbb{C} _N(\\mathcal{F})$ complexes (resp. $\\mathbb{C} _N(\\mathcal{C})$ complexes), consisting of all $\\mathbf{X} \\in \\mathbb{C} _N(\\mathcal{A} )$ such that $X^i \\in \\mathcal{F}$ (resp. $X^i \\in \\mathcal{C}$) for each $i$. \n\\item[(2)]The class of $\\mathcal{F}$-$N$-complex, that we denote by $\\widetilde{\\mathcal{F}}_N$, consisting of all $\\mathbf{X} \\in \\mathcal{E}_N$ such that ${\\rm{Z}}_{r}^{i}(\\mathbf{X}) \\in \\mathcal{F}$ for all $r,i$.\n\\item[(3)] The class of $\\mathcal{C}$-$N$-complex, that we denote by $\\widetilde{\\mathcal{C}}_N$, consisting of all $\\mathbf{X} \\in \\mathcal{E}_N$ such that ${\\rm{Z}}_{r}^{i}(\\mathbf{X}) \\in \\mathcal{C}$ for all $r,i$.\n\\item[(4)]The class of ${\\rm{dg}}$-$\\mathcal{F}$-$N$-complexes, that we denote by ${\\rm{dg}} \\widetilde{\\mathcal{F}}_N$, consisting of all $\\mathbf{X} \\in \\mathbb{C} _N(\\mathcal{F})$ such that ${\\rm{Hom}}_{\\mathbb{K} _{N}(\\mathcal{A})}(\\mathbf{X},\\mathbf{C})=0$ whenever $\\mathbf{C} \\in \\widetilde{\\mathcal{C}}_N$.\n\\item[(4)] The class of ${\\rm{dg}}$-$\\mathcal{C}$-$N$-complexes, that we denote by ${\\rm{dg}} \\widetilde{\\mathcal{C}}_N$, consisting of all $\\mathbf{X} \\in \\mathbb{C} _N(\\mathcal{C})$ such that ${\\rm{Hom}}_{\\mathbb{K} _{N}(\\mathcal{A})}(\\mathbf{F},\\mathbf{X})=0$ whenever $\\mathbf{F} \\in \\widetilde{\\mathcal{F}}_N$.\n\\item[(5)]The class ${ex}_N(\\mathcal{F})=\\mathbb{C} _N(\\mathcal{F})\\cap \\mathcal{E}_N$(resp. ${ex}_N(\\mathcal{C})=\\mathbb{C} _N(\\mathcal{C})\\cap \\mathcal{E}_N$.\n\\end{itemize}\n\\end{definition}\n\\begin{example}\nLet ${\\rm{Prj}\\mbox{-}} R$ be the category of projective objects in ${\\rm{Mod\\mbox{-}}} R$. Consider the cotorsion pair $({\\rm{Prj}\\mbox{-}} R, {\\rm{Mod\\mbox{-}}} R)$. Then the purpose of a dg-projective $N$-complex is an $N$-complex $\\mathbf{P}$ such that $P^i \\in {\\rm{Prj}\\mbox{-}} R$ and  ${\\rm{Hom}}_{\\mathbb{K} _{N}(R)}(\\mathbf{P},\\mathbf{E})=0$ for all $\\mathbf{E}\\in \\mathcal{E}_N$. This definition is compatible with the definition 3.20 in \\cite{IKM}.\n\\end{example}\nThe next corollary is contained in \\cite[Theorem 3.7]{YC}. Here we present short proof of it for our case.\n\\begin{corollary}\n\\label{corollary 3.7}\nLet $(\\mathcal{F} ,\\mathcal{C} )$ be a cotorsion pair in $\\mathcal{A} $. Then $(\\widetilde{\\mathcal{F} }_N,dg\\widetilde{\\mathcal{C} }_N)$ and $(dg\\widetilde{\\mathcal{F} }_N,\\widetilde{\\mathcal{C} }_N)$ are cotorsion pairs in $\\mathbb{C} _N(\\mathcal{A} )$.\n\\end{corollary} \n\\begin{proof}\nWe just prove one of the statements since the other is dual. If we set consider $\\mathcal{F} =\\mathcal{X} $ and $\\mathcal{C} =\\mathcal{Y} $\nas in proposition \\ref{prop 3.3}, then we can say that $(\\widetilde{\\mathcal{F} }_{\\mathcal{F} _N},(\\widetilde{\\mathcal{F} }_{\\mathcal{F} _N})^\\perp)$ is a cotorsion pair. But clearly $\\widetilde{\\mathcal{F} }_{\\mathcal{F} _N}=\\widetilde{\\mathcal{F} }_N$ and $(\\widetilde{\\mathcal{F} }_{\\mathcal{F} _N})^\\perp)=dg\\widetilde{\\mathcal{C} }_N$.\n\\end{proof}\nNow let $(\\mathcal{P} ,\\mathcal{A} )$ and $(\\mathcal{A} ,\\mathcal{I} )$ be the usual projective and injective cotorsion pairs, where $\\mathcal{P} $ is the class of projective, $\\mathcal{I} $ is the class of injective objects in $\\mathcal{A} $. Note that for any cotorsion pair $(\\mathcal{F} ,\\mathcal{C} )$ in $\\mathcal{A} $ we always have inclusions $\\mathcal{P} \\subseteq \\mathcal{F} $ and $\\mathcal{I} \\subseteq\\mathcal{C} $. The next corollary is contained in \\cite[Proposition 4.2]{YC}. In the following, we provide a brief proof of the case, with the difference that we can omit the hereditary condition on $(\\mathcal{F} ,\\mathcal{C} )$.\n\\begin{corollary}\n\\label{corollary 3.8}\nLet $(\\mathcal{F} ,\\mathcal{C} )$ be a cotorsion pair in $\\mathcal{A} $. Then $(ex\\widetilde{\\mathcal{F} }_N,(ex\\widetilde{\\mathcal{F} }_N)^\\perp)$ and $({}^\\perp(ex\\widetilde{\\mathcal{C} }_N),ex\\widetilde{\\mathcal{C} }_N)$ are cotorsion pairs in $\\mathbb{C} _N(\\mathcal{A} )$.\n\\end{corollary} \n\\begin{proof}\nWe just prove one of the statements since the other is dual. In order to use proposition \\ref{prop 3.3} we consider $(\\mathcal{F} ,\\mathcal{C} )$ and $(\\mathcal{A} ,\\mathcal{I} )$. Note that $\\mathcal{F} \\subseteq \\mathcal{A} $ so $(ex\\widetilde{\\mathcal{X} }_N,(ex\\widetilde{\\mathcal{X} }_N)^\\perp)$ is a cotorsion pair since clearly $ex\\widetilde{\\mathcal{X} }_N=\\widetilde{\\mathcal{X} }_{\\mathcal{A} _N}$.\n\\end{proof} \n\n\nFor the rest of this section we assume that $\\mathcal{G} $ is a concrete Grothendieck category as in subsection \\ref{subsect 01}. In the following we will prove that the induced cotorsion pairs in $\\mathbb{C} _N(\\mathcal{G} )$ as above are also complete.  \n\\begin{theorem}\n\\label{theorem 3.6}\nSuppose that $(\\mathcal{F} ,\\mathcal{C} )$ and $(\\mathcal{X} ,\\mathcal{Y} )$ are two cotorsion pairs in $\\mathcal{G} $ with $\\mathcal{F} \\subseteq\\mathcal{X} $ and the generator of $\\mathcal{G} $ is in $\\mathcal{F} $. If both $(\\mathcal{F} ,\\mathcal{C} )$ and $(\\mathcal{X} ,\\mathcal{Y} )$ are cogenerated by sets, then the induced pairs $( \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}, (\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N})^\\perp)$ and $( {}^\\perp(\\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N}), \\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N})$ are complete cotorsion pairs.\n\\end{theorem}\nWe will prove the theorem in two steps. First we show that $( \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}, (\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N})^\\perp)$ is a complete cotorsion pair.\n\\begin{proposition}\n\\label{Prop 0001}\nLet $(\\mathcal{F} ,\\mathcal{C} )$ and $(\\mathcal{X} ,\\mathcal{Y} )$ be two cotorsion pairs with $\\mathcal{F} \\subseteq\\mathcal{X} $ in $\\mathcal{G} $ such that the generator $G$ in $\\mathcal{G} $ is in $\\mathcal{F} $. If both  $(\\mathcal{F} ,\\mathcal{C} )$ and $(\\mathcal{X} ,\\mathcal{Y} )$ are cogenerated by sets, then so is the induced cotrsion pair $( \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}, (\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N})^\\perp)$ and so it is complete.\n\\end{proposition}\n\\begin{proof}\nBy Theorem \\ref{theorem 2.12} it is enough to show that each complex $\\mathbf{F}\\in\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$ is ${\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}}^\\kappa$-filtered (for some $\\kappa\\geq\\lambda$ regular uncountable) i.e. we construct a filtration $(\\mathbf{F}_\\alpha \\mid \\alpha\\leq\\sigma)$ for $\\mathbf{F}$ such that $\\mathbf{F}_{\\alpha+1}\/{\\mathbf{F}_\\alpha}\\in {\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}}^\\kappa$. \nLet $\\mathbf{F}=(F^i)\\in \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$. By definition $\\mathbf{F}$ is an $N$-exact complex with $F^i\\in \\mathcal{F} $ and ${\\rm{Z}}^i_r(\\mathbf{F})\\in \\mathcal{X} $ for $i\\in \\mathbb{Z} $ and $1\\leq r \\leq N-1$. Since we have $\\mathcal{F} \\subseteq \\mathcal{X} $, it is also ${\\rm{Z}}^i_{N}(\\mathbf{F}) = F^i\\in \\mathcal{X} $. By assumption ${\\rm{Z}}^i_r(\\mathbf{F})$ has $\\mathcal{X} ^\\kappa$- filtration $\\mathcal{M} _{i,r}=(M^{i,r}_\\alpha \\mid \\alpha\\leq \\sigma_{i,r})$ for each $i\\in \\mathbb{Z} $, $1\\leq r \\leq N$. Using Hill Lemma, we obtain the corresponding families $\\mathcal{H} ^{i,r}$ for these filtrations.\n\nNow, we recursively construct a filtration $(\\mathbf{F}_\\alpha\\in \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N} \\mid \\alpha\\leq\\sigma)$ for $\\mathbf{F}$ with the property that, for each $\\alpha<\\sigma, i\\in\\mathbb{Z} $ and $1\\leq r\\leq N$, the object ${\\rm{Z}}^i_r(\\mathbf{F}_\\alpha)$ belongs to $\\mathcal{H} ^{i,r}$. First, put $\\mathbf{F}_0 = 0$. If $\\alpha$ is a limit ordinal and $\\mathbf{F}_\\beta$ is already defined for each $\\beta<\\alpha$, we simply put $\\mathbf{F}_\\alpha = \\bigcup_{\\beta<\\alpha} \\mathbf{F}_\\beta$. This is again an $N$-exact complex and, by the properties of Hill families, we have  ${\\rm{Z}}^i_r(\\mathbf{F}_\\alpha)\\in\\mathcal{H} ^{i,r}$ for all $i\\in\\mathbb{Z} $ and $1\\leq r\\leq N$. We proceed to the crucial isolated step. Let $\\mathbf{F}_\\alpha$ be defined and assume that $\\mathbf{F}_\\alpha \\neq \\mathbf{F}$ (otherwise, we set $\\sigma = \\alpha$ and we are done). Put ${\\rm{G}}_0 = \\mathbf{F}_\\alpha$.\n\nFor each $i\\in\\mathbb{Z} $, fix some $M_0^i\\in\\mathcal{H} ^{i,N}$ such that $G_0^i\\subseteq M_0^i$, $M_0^i\/G_0^i$ is $\\kappa$-presentable and, if possible, $G_0^i\\subsetneq M_0^i$. Assuming that $M_n^i$ is defined for some nonnegative integer $n$ and all $i\\in\\mathbb{Z} $, and $M_n^i\/G_0^i$ is $\\kappa$-presentable, the objects $(M_n^i\\cap {\\rm{Z}}^i_r(\\mathbf{F}))\/{\\rm{Z}}^i_r(\\mathbf{F}_\\alpha)$ are $\\kappa$-presentable as well for all $1\\leq r\\leq N-1$. Hence we can find $Z_n^{i,r}\\in\\mathcal{H} ^{i,r}$, $1\\leq r < N$, such that $M_n^i\\cap {\\rm{Z}}^i_r(\\mathbf{F})\\subseteq Z_n^{i,r}$ and $Z_n^{i,r}\/{\\rm{Z}}^i_r(\\mathbf{F}_\\alpha)$ is again $\\kappa$-presentable. We define $M_{n+1}^i\\in\\mathcal{H} ^{i,N}$ in such a way that $M_n^i\\cup\\bigcup_{r=1}^{N-1} Z_n^{i,r}\\subseteq M_{n+1}^i$ and $M_{n+1}^i\/M_n^i$ is $\\kappa$-presentable. This is possible by the properties of the Hill family $\\mathcal{H} ^{i,N}$. Consequently, $M_{n+1}^i\/G_0^i$ is $\\kappa$-presentable. For each $i\\in\\mathbb{Z} $, put $M^i = \\bigcup_{n=0}^\\infty M_n^i$. Then $M^i\/G_0^i$ is $\\kappa$-presentable. Moreover, $M^i\\cap {\\rm{Z}}^i_r(\\mathbf{F}) = \\bigcup_{n=0}^\\infty Z_n^{i,r}\\in\\mathcal{H} ^{i,r}$ for each $i\\in\\mathbb{Z} $ and $1\\leq r\\leq N-1$ and $M^i = \\bigcup_{n=0}^\\infty M_n^i\\in\\mathcal{H} ^{i,N}$.\n\nNow, we use Lemma~\\ref{lemma2.14} to obtain an $N$-exact complex ${\\rm{G}}_1$ such that ${\\rm{G}}_0\\subseteq {\\rm{G}}_1\\subseteq \\mathbf{F}$, the quotient $G_1^i\/G_0^i$ is $\\kappa$-presentable and $M^i\\subseteq G_1^i$ for each $i\\in\\mathbb{Z} $. We go back to the beginning of the previous paragraph and repeat the process with ${\\rm{G}}_0$ replaced by ${\\rm{G}}_1$. Using Lemma~\\ref{lemma2.14}, we obtain ${\\rm{G}}_2$ and so on. Finally, we define $\\mathbf{F}_{\\alpha+1} = \\bigcup_{n=0}^\\infty {\\rm{G}}_n$. This is an $N$-exact complex and, for all $i\\in \\mathbb{Z} $, ${\\rm{Z}}^i_r(\\mathbf{F}_{\\alpha+1}) = F_{\\alpha+1}\\cap{\\rm{Z}}^i_r(\\mathbf{F})$ is the union of elements of the type $M^i\\cap {\\rm{Z}}^i_r(\\mathbf{F})\\in\\mathcal{H} ^{i,r}$; thus ${\\rm{Z}}^i_r(\\mathbf{F}_{\\alpha+1})$ is an element from $\\mathcal{H} ^{i,r}$ for all $i\\in\\mathbb{Z} $ and $1\\leq r\\leq N$. Moreover, $F^i_{\\alpha+1}\/F^i_\\alpha$ is $\\kappa$-presentable.\n\nThis finishes the construction of the filtration $(\\mathbf{F}_\\alpha \\mid \\alpha\\leq\\sigma)$. Finally, we observe that, for each $\\alpha<\\sigma$, the quotient $\\mathbf{F}_{\\alpha+1}\/\\mathbf{F}_\\alpha$ belongs to ${\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}}^\\kappa$: here ${\\rm{Z}}^i_r(\\mathbf{F}_{\\alpha+1})\/{\\rm{Z}}^i_r(\\mathbf{F}_\\alpha)\\in\\mathcal{X} $ since ${\\rm{Z}}^i_r(\\mathbf{F}_{\\alpha}), {\\rm{Z}}^i_r(\\mathbf{F}_{\\alpha+1})\\in\\mathcal{H} ^{i,r}$ for all $i\\in\\mathbb{Z} $ and $1\\leq r < N$.\n\nThe completeness of pair $( \\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}, (\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N})^\\perp)$ follows as \\cite[Corollary 6.6]{Hov02} because $\\widetilde{\\mathcal{F} }_{\\mathcal{X} _N}$ contains a generating set of $\\mathbb{C} _N(\\mathcal{G} )$.\n\\end{proof}\n\n\n\\begin{proposition}\n\\label{Prop 0002}\nLet $(\\mathcal{F} ,\\mathcal{C} )$ and $(\\mathcal{X} ,\\mathcal{Y} )$ be two cotorsion pairs with $\\mathcal{F} \\subseteq\\mathcal{X} $ in $\\mathcal{G} $ such that the generator $G$ in $\\mathcal{G} $ is in $\\mathcal{F} $. If both  $(\\mathcal{F} ,\\mathcal{C} )$ and $(\\mathcal{X} ,\\mathcal{Y} )$ are cogenerated by sets, then so is the induced cotrsion pair $( {}^\\perp(\\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N}), \\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N})$ and so it is complete.\n\\end{proposition}\n\\begin{proof}\nSuppose that $(\\mathcal{F} ,\\mathcal{C} )$ is cogenerated by a set $\\lbrace A_j\\,\\, |\\,\\, j\\in J\\rbrace$ and $(\\mathcal{X} ,\\mathcal{Y} )$ is cogenerated by the set $\\lbrace B_k\\,\\, |\\,\\, k\\in K\\rbrace$ . We claim that $({}^\\perp(\\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N}), \\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N})$ is cogenerated by \n\\begin{align*}\n\\mathcal{S}  &=\\lbrace D^i_r(G)|\\,\\, i\\in \\mathbb{Z} , \\,\\, 1\\leq r \\leq N-1\\rbrace \\cup \\lbrace D^i_r(A_j)|\\,\\, i\\in \\mathbb{Z} ,\\,\\, 1\\leq r \\leq N-1, \\,\\, j\\in J \\rbrace \n\\\\\n\\qquad &\\cup \\lbrace D^i_N(B_k)|\\,\\, i\\in \\mathbb{Z} ,\\,\\, 1\\leq r \\leq N-1, \\,\\, k\\in K \\rbrace\n\\end{align*}\n \nIn dual manner of the proposition \\ref{prop 3.3} we can prove that $D^i_r(F)\\in {}^\\perp(\\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N})$ whenever $F\\in \\mathcal{F} $ and $D^i_N(X)\\in {}^\\perp(\\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N})$ whenever $X\\in \\mathcal{X} $. So we have  $\\mathcal{S} \\subseteq {}^\\perp(\\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N})$. Thus $\\mathcal{S} ^\\perp\\supseteq {}^\\perp(\\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N})^\\perp=\\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N}$. \nConversely, let $\\mathbf{Y}\\in \\mathcal{S} ^\\perp$. First, we show that $\\mathbf{Y}$ is an $N$-exact complex. Consider the exact sequence $0 \\rightarrow D^{i+r-1}_r(G)\\rightarrow D^{i+N-1}_N(G)\\rightarrow D^{i+N-1}_{N-r}(G)\\rightarrow 0$. It induces an exact sequence\n$${\\rm{Hom}}_{\\mathbb{C} _N(\\mathcal{G} )}(D^{i+N-1}_N(G), \\mathbf{Y})\\rightarrow {\\rm{Hom}}_{\\mathbb{C} _N(\\mathcal{G} )}(D^{i+r-1}_r(G), \\mathbf{Y}) \\rightarrow {\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{G} )}(D^{i+N-1}_{N-r}(G), \\mathbf{Y}) $$\nBut ${\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{G} )}(D^{i+N-1}_{N-r}(G), \\mathbf{Y})=0$, since $\\mathbf{Y}\\in \\mathcal{S} ^\\perp$. Hence, by \\cite[Lemma 2.3]{YC} we can say that $\\mathbf{Y}$ is an $N$-exact complex. On the other hand, by lemma \\ref{lemma01} we have \n${\\rm{Ext}}^1_{\\mathcal{G} }(A_j, {\\rm{Z}}^i_r(\\mathbf{Y}))\\cong {\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{G} )}(D^{i+r-1}_{r}(A_j), \\mathbf{Y})=0$. This implies ${\\rm{Z}}^i_r(\\mathbf{Y})\\in \\mathcal{C} $ since $\\lbrace A_j\\,\\, |\\,\\, j\\in J\\rbrace$ cogenerates the cotorsion pair $(\\mathcal{F} ,\\mathcal{C} )$. Also ${\\rm{Ext}}_{\\mathcal{G} }^{1}(B_k,Y^i)\\cong {\\rm{Ext}}_{\\mathbb{C} _N(\\mathcal{G} )}^1(D_{N}^{i+N-1}(B),\\mathbf{Y})=0$ for all $k\\in K$. Thus $Y^i\\in \\mathcal{Y} $,  since $(\\mathcal{X} ,\\mathcal{Y} )$ is cogenerated by the set $\\lbrace B_k\\,\\, |\\,\\, k\\in K\\rbrace$ \n\nFinally, since $G$ generates $\\mathcal{G} $, the complexes $D^i_N(G)$ generates $\\mathbb{C} _N(\\mathcal{G} )$. Also $D^i_N(G)\\in {}^\\perp(\\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N})$ and so ${}^\\perp(\\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N})$ contains the generators $\\lbrace D^i_N(G)\\,\\,|\\,\\, i\\in \\mathbb{Z} \\rbrace$. So by \\cite[Corollary 6.6]{Hov02} we have the completeness of the pair $( {}^\\perp(\\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N}), \\widetilde{\\mathcal{Y} }_{\\mathcal{C} _N})$\n\\end{proof}\n\n\\begin{corollary}\n\\label{corollary 3.13}\nLet $(\\mathcal{F} ,\\mathcal{C} )$ be a cotorsion pairs in a concrete category $\\mathcal{G} $ as in subsection \\ref{subsect 01} and such that the generator of $\\mathcal{G} $ is in $\\mathcal{F} $. Then \n\\begin{itemize}\n\\item[(1)]$(\\widetilde{\\mathcal{F} }_N,dg\\widetilde{\\mathcal{C} }_N)$ and $(dg\\widetilde{\\mathcal{F} }_N,\\widetilde{\\mathcal{C} }_N)$\n\\item[(2)]$(ex\\widetilde{\\mathcal{F} }_N,(ex\\widetilde{\\mathcal{F} }_N)^\\perp)$ and $({}^\\perp(ex\\widetilde{\\mathcal{C} }_N),ex\\widetilde{\\mathcal{C} }_N)$\n\\end{itemize}\nare complete cotorsion pairs in $\\mathbb{C} _N(\\mathcal{A} )$.\n\\end{corollary}\n\\begin{proof}\nUsing the proof of corollary \\ref{corollary 3.7}, \\ref{corollary 3.8} and Theorem \\ref{theorem 3.6}.\n\\end{proof}\n\n\nNote that the previous results are improved versions of \\cite[Theorem 3.13]{YC} and\\cite[Proposition 4.8, 4.9]{YC}. Essentially we do not assume that the cotorsion pair $(\\mathcal{F} ,\\mathcal{C} )$ is complete hereditary.\n\n\n\\begin{example}\nLet $\\mathcal{Q}  co(\\mathbb{X})$ be the category of quasi-coherent sheaves on a scheme $\\mathbb{X}$. Then $\\mathcal{Q}  co(\\mathbb{X})$ is a Grothendieck category as \\ref{subsect 01}. Note that $U(F)=\\sqcup_{v\\in \\mathcal{V} }F(v)$, where $\\mathcal{V} $ is a fixed open affine cover of $\\mathbb{X}$. If we let $\\mathcal{F} $ be the class of all flat quasi-coherent sheaves, it is known that if $\\mathbb{X}$ is quasi-compact and semi-separated, then $\\mathcal{F} $ contains a generator of  $\\mathcal{Q}  co(\\mathbb{X})$. Moreover, by \\cite[Section 4]{EE} we follow that $(\\mathcal{F} ,\\mathcal{F} ^\\perp)$ is cogenerated by a set. So corollary \\ref{corollary 3.13} apply.\nAgain, consider the category of quasi-coherent sheaves and let $\\mathcal{F} $ be the\nclass of (non\u2013necessarily finite dimensional) vector bundles and the class of ``restricted'' Drinfeld\nvector bundles (see \\cite{EAPT} for notation and terminology) on suitable schemes. These classes are not\nin general closed under direct limits but we can proceed in the same way and apply corollary \\ref{corollary 3.13}.\n\\end{example}\n\n\\section{Applications}\n\\label{section 4}\n Let $R$ be an associative unitary ring. Let $\\mathbb{K} _N({\\rm{Flat}\\mbox{-}} R)$ be the homotopy category of $N$-complexes of flat $R$-modules, and let $\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)$ be the homotopy category of $N$-complexes of projective $R$-modules. In \\cite{BHN} the authors proved that $\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)$ is equivalent to $\\mathbb{K} ({\\rm{Prj}\\mbox{-}} \\mathbb{T} _{N-1}(R))$ whenever $R$ is a left coherent ring and $\\mathbb{T} _{N-1}(R)$ is the ring of triangular matrices of order $N-1$ with entries in $R$.\nThis equivalence allows us to study the properties of $\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)$ from $\\mathbb{K} ({\\rm{Prj}\\mbox{-}} \\mathbb{T} _{N-1}(R))$. For instance  $\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)$ is compactly generated whenever $R$ is a left coherent ring. There is a natural question and this is whether it is possible to introduce an $N$-complex version of \\cite[Theorem 0.1]{Nee10}, \\cite[Proposition 8.1]{Nee08}? The answer is not trivial, since we don not have such an equivalence for $\\mathbb{K} _N({\\rm{Flat}\\mbox{-}} R)$ and $\\mathbb{K} ({\\rm{Flat}\\mbox{-}} \\mathbb{T} _{N-1}(R))$. In this section we focus on particular homotopy categories and the existence of adjoint functor between them. First, we start with the following lemma.\n\n\\begin{lemma}\n\\label{lemma 41}\nLet $\\mathcal{G} $ be a Grothendieck category. Let $\\mathbf{X}$ and $\\mathbf{Y}$ be in $\\mathbb{C} _N(\\mathcal{G} )$. Given $f\\in {\\rm{Hom}}_{\\mathbb{C} _N(\\mathcal{G} )}(\\mathbf{X},\\mathbf{Y})$ an associated exact sequence $0\\rightarrow \\mathbf{Y}\\xrightarrow{u}C(f)\\rightarrow \\Sigma \\mathbf{X}\\rightarrow 0$. Then $u$ is split monomorphism in $\\mathbb{C} _N(\\mathcal{G} )$ if and only if it is split monomorphism in $\\mathbb{K} _N(\\mathcal{G} )$.\n\\end{lemma}\n\\begin{proof}\n``$\\Rightarrow$'' is clear. Conversely suppose that $\\mathbf{Y}\\xrightarrow{u}C(f)$ is split monomorphism in $\\mathbb{K} _N(\\mathcal{G} )$. So there is a morphism $r:C(f)\\rightarrow \\mathbf{Y}$ such that $ru\\sim 1_{\\mathbf{Y}}$. Let $t$ be the corresponding homotopy as in the definition \\ref{def 11}. Define $a:C(f)\\rightarrow \\mathbf{Y}$ by \n$$\n(y,x_1,x_2,...x_{N-1})\\longmapsto y+\\sum_{r=1}^{N-1}\\sum_{i=1}^{N-r}d_{\\mathbf{Y},\\lbrace N-i-r\\rbrace}^{n-(N-i-r)} t^{n+i+r-1} d_{\\mathbf{Y},\\lbrace i-1\\rbrace}^{n+r}f^{n+r}(x_r)+r^n(0,x_1,x_2,...x_{N-1})\n$$\nClearly $au=1_{\\mathbf{Y}}$. So it is enough to show that $a=(a^n)_{n\\in \\mathbb{Z} }$ is a morphism in $\\mathbb{C} _N(\\mathcal{G} )$, i.e $d^n_\\mathbf{Y} a^n=a^{n+1}d^n_{C(f)}$ for all $n\\in \\mathbb{Z} $. Given $(y, x_1,...,x_{N-1})\\in Y^n \\oplus \\coprod_{i=n+1}^{n+N-1} X^i$, so we need to show that \n\\begin{align*}\nd^n_{\\mathbf{Y}}(y)&+\\sum_{r=1}^{N-1}\\sum_{i=1}^{N-r}d_{\\mathbf{Y},\\lbrace N-i-r+1\\rbrace}^{n-(N-i-r)} t^{n+i+r-1} d_{\\mathbf{Y},\\lbrace i-1\\rbrace}^{n+r}f^{n+r}(x_r)+d^n_\\mathbf{Y} r^n(0,x_1,x_2,...x_{N-1})\\\\\n&= d^n_\\mathbf{Y}(y) + f^{n+1}(x_1)+ \\sum_{r=1}^{N-2}\\sum_{i=1}^{N-r}d_{\\mathbf{Y},\\lbrace N-i-r\\rbrace}^{n+1-(N-i-r)} t^{n+i+r} d_{\\mathbf{Y},\\lbrace i-1\\rbrace}^{n+r+1}f^{n+r+1}(x_{r+1})\\\\\n&\\qquad\\qquad - \\sum_{i=1}^{N-r}t^{n+N}f^{n+N}d^{n+i}_{\\mathbf{X},\\lbrace N-i\\rbrace}(x_i)+r^{n+1}(0,x_2,x_3,...x_{N-1},-\\sum_{i=1}^{N-r}d^{n+i}_{\\mathbf{X},\\lbrace N-i\\rbrace}(x_i))\n\\end{align*}\nCanceling the same terms from both side and using the fact that $f$ is a morphism in $\\mathbb{C} _N(\\mathcal{G} )$ and $ru\\sim 1_\\mathbf{Y}$, we are reduced to show that\n\\begin{align*}\n(1-r^{n+1}u^{n+1})(f^{n+1}(x_1))+d^n_\\mathbf{Y} r^n(0,x_1,x_2,...,x_{N-1})&=f^{n+1}(x_1) \\\\\n&+r^{n+1}(0,x_2,...,x_{N-1},-\\sum_{i=1}^{N-r}d^{n+i}_{\\mathbf{X},\\lbrace N-i\\rbrace}(x_i))\n\\end{align*}\nOr equivalently,\n$$\nr^{n+1}(f^{n+1}(x_1),0,...,0)=d^n_\\mathbf{Y} r^{n}(0,x_1,...,x_{N-1})+r^{n+1}(0,-x_2,...,-x_{N-1},\\sum_{i=1}^{N-r}d^{n+i}_{\\mathbf{X},\\lbrace N-i\\rbrace}(x_i))\n$$\nand this equation satisfies, Since \n$$d^n_\\mathbf{Y} r^{n}(0,x_1,...,x_{N-1})= r^{n+1}(f^{n+1}(x_1),x_2,...,x_{N-1},-\\sum_{i=1}^{N-r}d^{n+i}_{\\mathbf{X},\\lbrace N-i\\rbrace}(x_i))$$\n\\end{proof}\n\nThe idea of the proof of the following Theorem is taken from \\cite[Theorem 3.5]{EBIJR}. We provide here the argument for the reader's convenience.\n\n\\begin{theorem}\n\\label{theorem 42}\nLet $(\\mathcal{F} ,\\mathcal{C} )$ be a cotorsion pair in $\\mathbb{C} (\\mathcal{G} )$ such that $\\mathcal{F} $ is closed under taking suspensions. Then the embedding $\\mathbb{K} _N(\\mathcal{F} )\\rightarrow \\mathbb{K} _N(\\mathcal{G} )$ has a right adjoint.\n\\end{theorem}\n\\begin{proof}\nWe define right adjoint $T:\\mathbb{K} _N(\\mathcal{G} )\\rightarrow \\mathbb{K} _N(\\mathcal{F} )$ as follows\n\n\\textbf{On object:} Let $\\mathbf{X}\\in\\mathbb{C} _N(\\mathcal{G} )$. Consider an exact sequence $0\\rightarrow \\mathbf{C} \\rightarrow \\mathbf{F}\\rightarrow \\mathbf{X}\\rightarrow 0$ with $\\mathbf{F}\\in \\mathcal{F} $ and $\\mathbf{C}\\in \\mathcal{C} $. Then define $T(\\mathbf{X}):=\\mathbf{F}$.\n\n\\textbf{On Morphism:} Let $f:\\mathbf{X}\\rightarrow \\mathbf{X}'$ be a morphism in $\\mathbb{C} _N(\\mathcal{G} )$ and consider the following diagram:\n$$\\xymatrix{0 \\ar[r] & \\mathbf{C} \\ar[r]  & \\mathbf{F} \\ar[r]^{p}  & \\mathbf{X} \\ar[d]^{f}\\ar[r] & 0 \\\\ 0 \\ar[r] & \\mathbf{C}' \\ar[r] & \\mathbf{F}' \\ar[r]^{q} & \\mathbf{X}' \\ar[r]  & 0 }$$\nBut we have the exact sequence $${\\rm{Hom}}_{\\mathbb{C} _N(\\mathcal{G} )}(\\mathbf{F},\\mathbf{F}')\\rightarrow {\\rm{Hom}}_{\\mathbb{C} _N(\\mathcal{G} )}(\\mathbf{F},\\mathbf{X}')\\rightarrow {\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{G} )}(\\mathbf{F},\\mathbf{C}')=0$$\nHence there exists $g\\in {\\rm{Hom}}_{\\mathbb{C} _N(\\mathcal{G} )}(\\mathbf{F},\\mathbf{F}')$ such that $fp=qg$. So define $T(f):=g$. This definition is well defined up to homotopy. Indeed, if $f_1, f_2 :\\mathbf{X}\\rightarrow \\mathbf{X}'$ are two morphisms such that $f_1\\sim f_2$ and suppose that $T(f_1)=g_1$ and $T(f_2)=g_2$, then we claim $g_1\\sim g_2$. Since $f_1\\sim f_2$ we can say that $f_1 p\\sim f_2 p$ and therefore $f=(f_1 p-f_2 p) \\sim 0$. We show that $g=(g_1-g_2)\\sim 0$. To this point consider the following diagram:\n$$\\xymatrix{0 \\ar[r] & \\mathbf{F}' \\ar[r]^{i}\\ar[d]^{q}  & C(g)\\ar[r] \\ar@{-->}[d]^t  & \\Sigma \\mathbf{F} \\ar@{=}[d]\\ar[r] & 0 \\\\ 0 \\ar[r] & \\mathbf{X}' \\ar[r]^{j} & C(f) \\ar[r] & \\Sigma \\mathbf{F} \\ar[r]  & 0 }$$\nSince $f\\sim 0$, by \\cite[proposition 2.14]{YD} we get that the lower short exact sequence splits. Consider $r:C(f)\\rightarrow \\mathbf{X}'$. Since $\\mathbf{F}'\\rightarrow \\mathbf{X}'$ is an $\\mathcal{F} $-precover, then there exists $\\ell:C(g)\\rightarrow \\mathbf{F}'$ such that $rt=q\\ell$. We claim that $\\ell$ provides a retraction of $i:\\mathbf{F}'\\rightarrow C(g)$ in $\\mathbb{K} _N(\\mathcal{G} )$. For this, it is easy to check that $q(1_{\\mathbf{F}'}-\\ell i)=0$, So we can say that  $1_{\\mathbf{F}'}-\\ell i$ maps $\\mathbf{F}'$ into the kernel of $q$, that is, into $\\mathbf{C}'$. Again by \\cite[proposition 2.14]{YD} and using this fact ${\\rm{Ext}}^1_{\\mathbb{C} _N(\\mathcal{G} )}(\\Sigma \\mathbf{F}',\\mathbf{C}')=0$ we can say that $1_{\\mathbf{F}'}-\\ell i$ is homotopic to $0$. So $\\ell i\\sim 1_{\\mathbf{F}'}$, i.e. $\\ell$ provides a retraction of $i:\\mathbf{F}'\\rightarrow C(g)$ in $\\mathbb{K} _N(\\mathcal{G} )$. By Lemma \\ref{lemma 41} $\\mathbf{F}'\\rightarrow C(g)$ is split monomorphism in $\\mathbb{C} _N(\\mathcal{G} )$, hence $0\\rightarrow \\mathbf{F}' \\rightarrow C(g) \\rightarrow \\Sigma \\mathbf{F}\\rightarrow 0$ is split exact. Therefore by \\cite[proposition 2.14]{YD}, we get that $g\\sim 0$.\n\nClearly we see that if $g_1\\sim g_2$ then $f_1\\sim f_2$. Hence  $$\\psi:{\\rm{Hom}}_{\\mathbb{K} _N(\\mathcal{F} )}(\\mathbf{F}'',T(\\mathbf{X}))\\rightarrow {\\rm{Hom}}_{\\mathbb{K} _N(\\mathcal{G} )}(\\mathbf{F}'',\\mathbf{X})$$ is injective. Clearly $\\psi$ is surjective and so it is bijective.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma 43}($N$-complex version of Neeman's result \\cite[Theorem 3.2]{Nee10})\nLet $R$ be a ring. The inclusion $i:\\mathbb{K} _N({\\rm{Flat}\\mbox{-}} R)\\rightarrow K_N({\\rm{Mod\\mbox{-}}} R)$ has a right adjoint functor.\n\\end{lemma}\n \\begin{proof}\n Consider the complete cotorsion pair $({\\rm{Flat}\\mbox{-}} R, {{\\rm{Flat}\\mbox{-}} R}^\\perp)$. By  \\cite[Proposition 4.4]{YC} we have a complete cotorsion pair $(\\mathbb{C} _N({\\rm{Flat}\\mbox{-}} R), \\mathbb{C} _N({\\rm{Flat}\\mbox{-}} R)^\\perp)$ in $\\mathbb{C} _N({\\rm{Mod\\mbox{-}}} R)$. Since $\\mathbb{C} _N({\\rm{Flat}\\mbox{-}} R)$ is closed under taking suspensions then by Theorem \\ref{theorem 42} $\\mathbb{K} _N({\\rm{Flat}\\mbox{-}} R)\\rightarrow \\mathbb{K} _N({\\rm{Mod\\mbox{-}}} R)$ has right adjoint functor $i^\\ast:\\mathbb{K} _N({\\rm{Mod\\mbox{-}}} R)\\rightarrow \\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)$.\n \\end{proof}\n\n\\begin{lemma}\n\\label{lemma 43}($N$-complex version of Neeman's result \\cite[Proposition 8.1]{Nee08})\nThe natural inclusion $j_{!}:\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)\\rightarrow K_N({\\rm{Flat}\\mbox{-}} R)$ has a right adjoint functor.\n\\end{lemma}\n \\begin{proof}\n Consider the complete cotorsion pair $({\\rm{Prj}\\mbox{-}} R, {\\rm{Mod\\mbox{-}}} R)$. By  \\cite[Proposition 4.4]{YC} we have a complete cotorsion pair $(\\mathbb{C} _N({\\rm{Prj}\\mbox{-}} R), \\mathbb{C} _N({\\rm{Prj}\\mbox{-}} R)^\\perp)$ in $\\mathbb{C} _N({\\rm{Mod\\mbox{-}}} R)$. Since $\\mathbb{C} _N({\\rm{Prj}\\mbox{-}} R)$ is closed under taking suspensions then by Theorem \\ref{theorem 42} $\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)\\rightarrow \\mathbb{K} _N({\\rm{Mod\\mbox{-}}} R)$ has right adjoint functor $j:\\mathbb{K} _N({\\rm{Mod\\mbox{-}}} R)\\rightarrow \\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)$. Then the natural inclusion $j_{!}:\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)\\rightarrow \\mathbb{K} _N({\\rm{Flat}\\mbox{-}} R)$ has a right adjoint $j^\\ast={j_{!}}|_{\\mathbb{K} _N({\\rm{Flat}\\mbox{-}} R)}$. \n \\end{proof}\n \n\\begin{lemma}\\label{lemma 44}($N$-complex version of Neeman's result \\cite[Theorem 0.1]{Nee10}) The functor $j^\\ast:\\mathbb{K} ({\\rm{Flat}\\mbox{-}} R)\\rightarrow \\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)$ has a right adjoint functor. \n\\end{lemma}\n\\begin{proof}\nThe functor $j_{!}:\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)\\rightarrow K_N({\\rm{Flat}\\mbox{-}} R)$ is fully faithful and by Lemma \\ref{lemma 43} has a right adjoint $j^\\ast$. Formal nonsense tell us that the right adjoint functor $j^\\ast:\\mathbb{K} _N({\\rm{Flat}\\mbox{-}} R)\\rightarrow \\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)$ is a Verdier quotient. The same formal nonsense also tell us that the right adjoint of Verdier quotient is fully faithful. By \\cite[Remark 2.12]{Nee08} this adjoint functor identifies $\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)$ with the Verdier quotient map\n$$\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)\\rightarrow \\mathbb{K} _N({\\rm{Flat}\\mbox{-}} R)\/\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)^\\perp$$\nwhere \n$$\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)^\\perp=\\lbrace Y\\in \\mathbb{K} _N({\\rm{Flat}\\mbox{-}} R)\\, | \\, {\\rm{Hom}}(j_{!}X,Y)=0\\,:\\, \\forall X\\in \\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)\\rbrace$$\nBut we can say thet \n\\begin{equation}\n\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)^\\perp\\xrightarrow{\\imath} \\mathbb{K} _N({\\rm{Flat}\\mbox{-}} R)\\xrightarrow{j^\\ast}\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R) \\label{eq-00}\n\\end{equation}\nis an quotient sequence of triangulated functor (see,the definitions in \\cite[chapter 2, pg.15]{Mur}). \nBut clearly, $\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)^\\perp$ concides with $\\mathbb{K} _N(\\widetilde{{\\rm{Flat}\\mbox{-}} R})$ (see the Definition \\ref{def 35} and \\cite[Fact 2.14]{Nee08}). Now, by Corollary \\ref{corollary 3.13} $(\\widetilde{{\\rm{Flat}\\mbox{-}} R}_N,\\widetilde{{\\rm{Flat}\\mbox{-}} R}^\\perp_N)$ is a complete cotorsion pair. So by Theorem \\ref{theorem 42} $\\mathbb{K} _N({\\rm{Prj}\\mbox{-}} R)^\\perp=\\mathbb{K} _N(\\widetilde{{\\rm{Flat}\\mbox{-}} R})\\rightarrow \\mathbb{K} _N({\\rm{Flat}\\mbox{-}} R)$ admits a right adjoint functor. So we can say that the sequence \\ref{eq-00} is a localization sequence. Hence by \\cite[Lemma 2.3]{Mur} $j^\\ast$ has a right adjoint.\n\n\\end{proof}\n\n\\section*{Acknowledgments}\n\nI would like to thank Jan \\v{S}aroch for his interest and for his pivotal role in proving the crucial Proposition \\ref{Prop 0001}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nThe main aim of this paper is to introduce a \\emph{continuous} positive operator valued measure (POVM) \\cite{NCh:00,AlFa:01}, which is \\emph{equivariant} under the symmetry group $SO(3,\\mathbb{R})$ of the qubit state space, and investigate the properties of the qubit state estimation protocol based on the generalized measurement defined by this POVM. There are several similar qubit state tomography protocols in the literature \\cite{PHSzSz:06,PHM:07} operating with cleverly chosen \\emph{discrete} measurements, which respect only a restricted, discrete group of symmetries. Probably the most famous of them is the ``minimal qubit tomography'' protocol \\cite{MinQubit:04}. I found it preferable to be a bit didactic here and to insert a brief account on these discrete qubit tomography protocols, for better comparability of the methods and results, and for the convenience of the reader. Thus the second section together with this introduction summarizes known results, while the new investigations are put in section~\\ref{s:contmeas}. The properties of the different protocols are compared in the concluding section.\n\nIn quantum information theory \\cite{NCh:00} the term \\emph{quantum bit} (or \\emph{qubit}) refers to the simplest nontrivial quantummechanical system, which has only two independent pure states. The Hilbert space of the qubit is $\\mathbb{C}^2$, and its possible states are described by the $2\\times 2$ density matrices $\\rho$, which can be decomposed in terms of the three Pauli matrices\n\\begin{equation}\\label{e:Pauli}\n\\begin{aligned}\n\\sigma_x &= \\begin{bmatrix} 0 &1 \\\\ 1 &0\\end{bmatrix},&\n\\sigma_y &= \\begin{bmatrix} 0 &-i \\\\ i &0\\end{bmatrix},\n\\\\\n\\sigma_z &= \\begin{bmatrix} 1 &0 \\\\ 0 &-1\\end{bmatrix},&\n\\bm{\\sigma} &=\\begin{bmatrix}\\sigma_x\\\\ \\sigma_y\\\\ \\sigma_z\\end{bmatrix}\n\\end{aligned}\n\\end{equation}\nin the following way:\n\\begin{equation}\\label{e:rhor}\n\\begin{aligned}\n\\rho(\\mathbf{r})&= \\frac{1}{2}(I+\\mathbf{r} \\bm{\\sigma}) =\\frac{1}{2}\n\\begin{bmatrix} 1+z& x-iy\\\\ x+iy& 1-z \\end{bmatrix},\n\\\\\n\\mathbf{r}&= \\begin{bmatrix}x\\\\y\\\\z\\end{bmatrix} \\in \\mathbb{R}^3,\n\\qquad\\qquad\n|\\mathbf{r}| =r \\le 1.\n\\end{aligned}\n\\end{equation}\n(The condition $r\\le 1$ ensures that $\\rho(\\mathbf{r}) \\ge 0$.) It means that the possible states of the qubit are in one-to-one correspondence with the points of the unit ball in $\\mathbb{R}^3$, which is conventionally called the \\emph{Bloch ball}. The surface of the ball, the \\emph{Bloch sphere} ($r=1$) represents the pure states, i.e., rank one projectors. The mapping $\\mathbf{r} \\mapsto \\rho(\\mathbf{r})$ respects convex combination (or ``averaging''), so\n\\begin{align}\\label{e:rhoav}\n\\rho(\\alpha \\mathbf{r}_1 +\\beta \\mathbf{r}_2 ) &=\\alpha \\rho(\\mathbf{r}_1) + \\beta \\rho(\\mathbf{r}_2);&\n\\langle \\rho(\\mathbf{r}) \\rangle &=\\rho(\\langle \\mathbf{r} \\rangle)\n\\end{align}\nprovided that $\\alpha +\\beta =1$.\n\nThe state space of the qubit has a nontrivial symmetry group determined by the automorphism group of its event lattice. In terms of density matrices the symmetry transformations are conjugations by unitaries, while in the Bloch ball picture the symmetries correspond to orthogonal rotations of the Bloch ball. The unitary group $SU(2,\\mathbb{C})$ is a two-fold covering of the orthogonal group $SO(3,\\mathbb{R})$, so they both have the same Lie-algebra. The structure of this Lie-algebra is reflected in the multiplication relations of the Pauli matrices, which we shall use in the sequel:\n\\begin{equation}\\label{e:sigma}\n\\begin{aligned}\n\\sigma_x^2 &=\\sigma_y^2 =\\sigma_z^2 =I,&\n\\sigma_x \\sigma_y &=-\\sigma_y \\sigma_x =i \\sigma_z,\\\\\n\\sigma_y \\sigma_z &=-\\sigma_z \\sigma_y =i \\sigma_x,&\n\\sigma_z \\sigma_x &=-\\sigma_x \\sigma_z =i \\sigma_y.\n\\end{aligned}\n\\end{equation}\n\nThe principal problem of quantum state estimation, or quantum tomography is to give an accurate estimation $\\rho'$ for an unknown qubit state $\\rho_0$ by performing certain quantum measurements on multiple replicas of the unknown quantum bit. In the present approach it is essential that all the replicas are in the same state $\\rho_0$, only one measurement is performed on each single replica, and the choice of the measurement does not depend on the previous results. Because of the inherent probabilistic nature of quantum mechanics the results of the measurements as well as the estimation $\\rho'$ itself are random variables. The estimation is \\emph{unbiased} if $\\langle \\rho' \\rangle =\\rho_0$, where $\\langle \\cdot \\rangle$ designates the expectation value taken over the possible outcomes of the measurements. The accuracy of the estimator $\\rho'$ is characterized by its variance $\\big\\langle d^2(\\rho',\\langle\\rho' \\rangle) \\big\\rangle$, where $d$ is an appropriate distance on the set of density matrices. A usual choice is the distance $d^2(\\rho_1,\\rho_2) =\\Tr(\\rho_2 -\\rho_1)^2$ based on the Hilbert-Schmidt norm for selfadjoint matrices.\n\nIt is worth noting that quantum state estimation is a much more delicate  problem than its classical counterpart because of two main reasons: $i)$~in quantum mechanics two quantities usually cannot be measured at the same time with arbitrary high precision; and $ii)$~measurements destroy the original state of the system. That is why in quantum tomography different, carefully chosen protocols exist for the measurements applied for the estimation of the unknown state.\n\nIn the second section we revisit three existing protocols and their basic properties. In all cases the measurements have a finite set of possible outcomes. The first protocol is based on von~Neumann (projective) spin measurements in three orthogonal directions, thus there are three different measurements, each having two possible outcomes. In the second protocol these three projective measurements are put together to obtain a (non-projective) measurement with six possible outcomes, based on a positive operator valued measure (POVM). The third one, the so called ``minimal state tomography'' protocol \\cite{MinQubit:04} is very much alike the previous protocol, but the number of possible outcomes of the POVM measurement is reduced to four, which is a lower bound in qubit state tomography.\n\nIn these three protocols the maximum likelihood method is applied to obtain the estimator $\\rho'$. Unfortunately in certain cases the likelihood function may take its maximum value outside the Bloch-ball, what makes the further exact analysis rather complicated \\cite{PHM:07}. Disregarding this fact, we show that the ``unrestricted'' estimator is unbiased (for the first and third protocol) or asymptotically unbiased (for the second protocol), and we calculate the variance of the (unrestricted) estimator in all cases. The variance goes to zero as the number $N$ of measurements is increased, what justifies that for large $N$ and mixed states the unrestricted and exact estimators are essentially the same. For pure states, however, the unrestricted estimator is not a good choice. Another principal disadvantage of these discrete protocols is the fact that they do not respect the whole symmetry group of the qubit.\n\nThis last defect is rectified in the third section, where we present a new protocol for qubit tomography based on a POVM, which is supported on the Bloch sphere, and \\emph{equivariant} under the symmetry group of the state space. An interesting novelty is that the POVM applied here is \\emph{continuous}, i.e., the corresponding measurement has an infinite number of possible outcomes, namely all the pure states. Although the maximum likelihood estimator cannot be explicitly constructed, we present another simple unbiased estimator and calculate its variance.\n\nWe conclude by comparing the results obtained for the different discrete and continuous qubit state estimation protocols.\n\n\\section{Discrete measurements}\nOriginally von Neumann defined quantum measurement as choosing one out of a set of pairwise complementary events which form a complete system \\cite{JvN:55}. Translating it into an algebraic language, a von~Neumann type (or projective) measurement is defined by a complete set of orthogonal projections $\\{ P_s \\}_{s\\in S}$. Here $S$ is an appropriate index set, and the projections satisfy the relations\n\\begin{align}\\label{e:vNm}\n\\sum_{s\\in S} P_s &=I,&\nP_s &=P_s^*,&\nP_s P_r &=\\delta_{s,r} P_s.\n\\end{align}\nIn the state $\\rho$ the probability that the measurement results in the event $P_s$ is $p_s =\\Tr (P_s \\rho)$. Usually it is convenient to ``label'' the possible outcomes, i.e., the index set $S$ by real numbers $a_s$ and define a selfadjoint operator $A= \\sum_{s\\in S} a_s P_s$ by its spectral decomposition. In this case we interpret $A$ as an \\emph{observable}, the $a_s$'s are the possible values of the observable (measurement), and the expectation value of $A$ is given by the well known formula $\\Tr (A\\rho)$. But the essential part of the von~Neumann type measurement is the orthogonal decomposition~\\eqref{e:vNm} of unity. Sometimes there is no natural way (or need) for the embedding $S\\stackrel{\\subset}{\\to} \\mathbb{R}$. In this case we can still speak about the probabilities $p_s$, which give a classical probability distribution on $S$, but (without further structure on $S$) the expectation value of the measurement has no meaning.\n\nThis scheme can be generalized to the so called \\emph{positive operator valued measure} (POVM) \\cite{NCh:00,AlFa:01}. In this case the identity is decomposed into the sum of (arbitrary) positive operators:\n\\begin{align}\n\\sum_{s\\in S} Q_s &= I,&\nQ_s &\\ge 0.\n\\end{align}\nThe possible outcomes of the measurement are labelled by the index set $S$, and the probability of the outcome $s$ in the state $\\rho$ is $p_s =\\Tr(Q_s \\rho)$. As in the previous case, the $p_s$'s define a classical probability distribution on $S$.\n\nThe POVM measurements are also called \\emph{weak measurements}, particularly if the investigated system is coupled to another system and the POVM measurement is obtained from a projective measurement performed on the composite system \\cite{NCh:00,AlFa:01}.\n\nIn this section we deal with \\emph{discrete measurements}, what means that the index set $S$ is finite. The continuous case is conceptually the same, only technically more difficult. We turn to this in section~\\ref{s:contmeas}.\n\n\\subsection{Orthogonal spin measurements I. --- Projective case}\nIn this protocol three different von~Neumann spin measurements are performed in three orthogonal directions (see figure~\\ref{f:sp}), so the projections are:\n\\begin{equation}\\label{e:Pxyzpm}\n\\begin{aligned}\nP_x^{\\pm} &=\\rho(\\pm\\mathbf{x})=\\frac{I\\pm\\sigma_x}{2},&\nP_y^{\\pm} &=\\rho(\\pm\\mathbf{y})=\\frac{I\\pm\\sigma_y}{2},\\\\\nP_z^{\\pm} &=\\rho(\\pm\\mathbf{z})=\\frac{I\\pm\\sigma_z}{2},&&\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbf{x}$, $\\mathbf{y}$ and $\\mathbf{z}$ denote the unit vectors in the three orthogonal directions.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.4]{spins.eps}\n\\end{center}\n\\caption{\\label{f:sp} The six projections belonging to the spin measurements in three orthogonal directions in the Bloch ball.}\n\\end{figure}\n\nLet $N_x= N_x^+ +N_x^-$, $N_y =N_y^+ +N_y^-$ and $N_z =N_z^+ +N_z^-$ be the number of spin measurements performed in the $x$, $y$ and $z$ direction, where $N_k^+$ is the number of $P_k^+$ results, and $N_k^-$ is the number of $P_k^-$ results ($k= x,y,z$). Furthermore, let $\\rho_0 =\\rho(\\mathbf{r}_0) =\\frac{1}{2}(I+\\mathbf{r}_0 \\bm{\\sigma})$ be the unknown true state of the qubit, and let $\\rho' =\\rho(\\mathbf{r}') =\\frac{1}{2}(I+\\mathbf{r}' \\bm{\\sigma})$ denote its maximum likelihood estimator.\n\nThe probabilities of the different measurement outcomes in the state $\\rho_0$ are\n\\begin{equation}\\label{e:ppmxz}\n\\begin{aligned}\np_x^{\\pm} &= \\Tr(\\rho_0 P_x^{\\pm}) =\\frac{1\\pm x_0}{2},&&\n\\\\\np_y^{\\pm} &=\\frac{1\\pm y_0}{2},&\np_z^{\\pm} &=\\frac{1\\pm z_0}{2},\n\\end{aligned}\n\\end{equation}\nso the likelihood function $\\mathcal L(\\mathbf{r}_0)$ for a given measurement statistics $N_k^{\\pm}$ is\n\\begin{multline}\\label{e:L}\n\\mathcal L(\\mathbf{r}_0) = \n\\binom{N_x}{N_x^+} (p_x^+)^{N_x^+} (p_x^-)^{N_x^-}\\\\\n\\times \\binom{N_y}{N_y^+} (p_y^+)^{N_y^+} (p_y^-)^{N_y^-}\n\\binom{N_z}{N_z^+} (p_z^+)^{N_z^+} (p_z^-)^{N_z^-}.\n\\end{multline}\nVarying the argument of the likelihood function, it takes the maximum value either on the boundary of the Bloch ball, or at the zero of its gradient:\n\\begin{equation}\n\\grad \\mathcal L(\\mathbf{r}') = \\mathcal L(\\mathbf{r}')\n\\begin{bmatrix}\n\\frac{N_x^+}{1+x'} -\\frac{N_x^-}{1-x'}\\\\\n\\frac{N_y^+}{1+y'} -\\frac{N_y^-}{1-y'}\\\\\n\\frac{N_z^+}{1+z'} -\\frac{N_z^-}{1-z'}\n\\end{bmatrix}\n=0\n\\end{equation}\nwhich yields\n\\begin{equation}\\label{e:rv}\n\\mathbf{r}' = \n\\begin{bmatrix}\n\\frac{N_x^+ -N_x^-}{N_x}\\\\\n\\frac{N_y^+ -N_y^-}{N_y}\\\\\n\\frac{N_z^+ -N_z^-}{N_z}\n\\end{bmatrix}.\n\\end{equation}\n\nA principal problem with this ``unrestricted'' estimator is that it may fall out of the Bloch ball. In this case either the true maximum place should be found on the boundary of the Bloch ball, or $\\mathbf{r}'$ should be normed in some other way, e.g., by dividing it by $|\\mathbf{r}'|$. For the moment we disregard this problem and continue as if $\\mathbf{r}'$ were a true estimator.\n\nUsing the binomial theorem we obtain that\n\\begin{equation}\\label{e:avN+}\n\\begin{split}\n\\Big\\langle \\frac{N^+}{N} \\Big\\rangle &=\n\\sum_{N^+ =0}^N \\frac{N^+}{N}\\binom{N}{N^+} {p^+}^{N^+} {p^-}^{N^-} \\\\\n&= \\sum_{N^+ =1}^N \\binom{N-1}{N^+ -1} p^+ {p^+}^{N^+ -1} {p^-}^{N^-} =p^+\n\\end{split}\n\\end{equation}\nin any of the three directions, and applying this and the previous equations \\eqref{e:ppmxz}, \\eqref{e:L}, \\eqref{e:rv} a straightforward calculation shows that the estimator $\\mathbf{r}'$ is unbiased:\n\\begin{equation}\n\\langle\\mathbf{r}'\\rangle =\\sum_{N_x^+=0}^{N_x} \\sum_{N_y^+=0}^{N_y} \\sum_{N_z^+=0}^{N_z}\n\\mathbf{r}' \\mathcal L(\\mathbf{r}_0) = \\mathbf{r}_0.\n\\end{equation}\n(The calculation can be carried out separately in the three spin directions.)\n\nNow let us determine the variance of the unrestricted estimator. Again a simple application of the binomial theorem shows that\n\\begin{multline}\\label{e:avN+2}\n\\Big\\langle \\frac{N^+ (N^+-1)}{N (N-1)} \\Big\\rangle =\n\\\\\n=\\sum_{N^+ =2}^N \\binom{N-2}{N^+ -2} {p^+}^2 {p^+}^{N^+ -2} {p^-}^{N^-} ={p^+}^2\n\\end{multline}\nin any of the three spin directions. The  distance square between the expectation value $\\langle \\rho' \\rangle =\\rho_0$ and the estimator is\n\\begin{equation}\\label{e:d2}\nd^2(\\rho',\\langle \\rho' \\rangle) =\\Tr(\\rho'-\\langle\\rho'\\rangle)^2 =\n\\frac{1}{4} \\Tr(\\Delta\\mathbf{r} \\bm{\\sigma})^2 = \\frac{1}{2}\\Delta r^2.\n\\end{equation}\n(Here $\\Delta \\mathbf{r}$ is a shorthand notation for the vector $\\mathbf{r}' -\\langle \\mathbf{r}' \\rangle$, and $\\Delta r$ is its absolute value.) Again the three directions decouple, what simplifies the calculations. The average of $\\Delta x^2$ can be determined with the help of \\eqref{e:avN+2}:\n\\begin{equation}\n\\langle \\Delta x^2 \\rangle = \\langle x'^2 \\rangle -\\langle x'\\rangle^2 =\n\\Big\\langle \\frac{(2N_x^+ -N_x)^2}{N_x^2} \\Big\\rangle -x_0^2 =\n\\frac{1-x_0^2}{N_x}.\n\\end{equation}\nIt means that in the states $\\rho_0 =P_x^{\\pm}$ the variance in the $x$ direction is zero, as it is expected. The total variance is obtained by summing similar expressions for the $x$, $y$ and $z$ direction:\n\\begin{equation}\\label{e:V3spin}\n\\begin{aligned}\nV_{\\ast} &=\n\\langle d^2(\\rho',\\rho_0)\\rangle =\n\\frac{1}{2} \\langle \\Delta x^2 +\\Delta y^2 + \\Delta z^2 \\rangle\n\\\\\n&=\\frac{1-x_0^2}{2N_x} +\\frac{1-y_0^2}{2N_x} +\\frac{1-z_0^2}{2N_z}.\n\\end{aligned}\n\\end{equation}\n\nIf we assume that $N_x =N_y =N_z =\\frac{N}{3}$, where $N =N_x +N_y +N_z$ is the total number of measurements, then the variance has the form\n\\begin{equation}\\label{e:V3}\nV_{\\ast} = \\frac{9-3r_0^2}{2N}.\n\\end{equation}\n\nThe variance tends to zero as the number of measurements goes to infinity, what justifies that for large $N$ and for mixed states the unrestricted estimator \\eqref{e:rv} practically coincides with the true maximum likelihood estimator.\n\n\\subsection{Orthogonal spin measurements II. --- POVM case \\label{ss:hexqubt}}\nThis protocol is also based on the six projections $P_{x,y,z}^{\\pm}$ appearing in the spectral decomposition of the three orthogonal spin operators (see equation~\\eqref{e:Pxyzpm}), but this time we form a single POVM out of them, by decomposing the unity in the following way:\n\\begin{equation}\nI =Q_x^+ +Q_x^- +Q_y^+ +Q_y^- +Q_z^+ +Q_z^- ,\n\\end{equation}\nwhere $Q_k^{\\pm} = \\frac{1}{3} P_k^{\\pm}$ for $k \\in \\{x,y,z\\}$ [see equation~\\eqref{e:Pxyzpm}]. Performing this POVM measurement on each replica of the qubit, the six different results are obtained with the probabilities\n\\begin{equation}\n\\begin{aligned}\nq_x^{\\pm} &=\\Tr(\\rho_0 Q_x^{\\pm}) =\\frac{1\\pm x_0}{6},&&\n\\\\\nq_y^{\\pm} &=\\frac{1\\pm y_0}{6},&\nq_z^{\\pm} &=\\frac{1\\pm z_0}{6},\n\\end{aligned}\n\\end{equation}\nprovided that the qubit is in the state $\\rho_0 =\\rho(\\mathbf{r}_0)$.\n\nLet $N_k =N_k^+ +N_k^-$ (for $k\\in \\{ x,y,z \\}$) be the number of measurements with any of the two possible results in the $k$ direction, and let $N=N_x +N_y +N_z$ denote the total number of measurements. The probability of a given measurement statistics $\\{ N_k^{\\pm} \\}$ in the state $\\rho_0$ is described by the multinomial distribution:\n\\begin{equation}\n\\mathcal L(\\mathbf{r}_0) =\nN! \\prod_{k\\in \\{x,y,z\\}} \\frac{(q_k^+)^{N_k^+} (q_k^-)^{N_k^-}}{N_k^+ ! N_k^- !}.\n\\end{equation}\n\nThe unrestricted maximum likelihood estimator $\\mathbf{r}'$ is obtained again by determining the zero of the gradient of $\\mathcal L(\\mathbf{r}')$, and we get formally the same expression~\\eqref{e:rv} as in the previous subsection. Now, however, an unavoidable problem is the fact that $N_x$, $N_y$ or $N_z$ may be zero with positive probability! If it happens then we have no information at all about certain component(s) of the Bloch vector $\\mathbf{r}'$, and the division in \\eqref{e:rv} is meaningless. In this case the most natural thing is to use the estimation $0$ for the appropriate component of $\\mathbf{r}'$. (We have to take care of this in the evaluation of the expectation values.) Of course this `ad hoc' choice will bias the estimator towards zero. In addition, the estimator may fall out of the Bloch ball; we disregard it in the followings.\n\nApplying the multinomial theorem it is easy to show that \n\\begin{widetext}\n\\begin{multline}\\label{e:avNx+}\n\\langle x' \\rangle =\n\\Big\\langle \\frac{N_x^+-N_x^-}{N_x} \\Big\\rangle_{N_x \\ge 1} =\n\\sum_{\\{N_k^{\\pm}\\}, N_x \\ge 1} \\frac{N_x^+ -N_x^-}{N_x} \\mathcal L(\\mathbf{r}_0) \n=\\sum_{N_x =1}^N \\binom{N}{N_x} (1-q_x)^{N-N_x} \\\\\n\\times \\sum_{N_x^+ =0}^{N_x} \n\\frac{N_x^+ -N_x^-}{N_x} \\binom{N_x}{N_x^+} (q_x^+)^{N_x^+} (q_x^-)^{N_x^-}\n=\\frac{q_x^+ -q_x^-}{q_x} \\sum_{N_x =1}^N \\binom{N}{N_x} (1-q_x)^{N-N_x} q_x^{N_x}\n=x_0 \\bigg(1-\\Big(\\frac{2}{3}\\Big)^N \\bigg).\n\\end{multline}\n\\end{widetext}\n(Here $q_x =q_x^+ +q_x^- =\\frac{1}{3}$, independently of the true state $\\rho_0$.) Similar expressions are valid in the other directions, and it follows that the estimator $\\mathbf{r}'$ is biased, but asymptotically unbiased:\n\\begin{equation}\\label{e:rbias}\n\\langle \\mathbf{r}' \\rangle =\\mathbf{r}_0 \\bigg(1-\\Big(\\frac{2}{3}\\Big)^N \\bigg).\n\\end{equation}\nNote that the factor $1-(2\/3)^N$ is exactly the probability that none of the $N$ measurement results lies in a particular coordinate direction.\n\nTo determine the variance of the estimator we need averages of the following type:\n\\begin{widetext}\n\\begin{equation}\\label{e:avNx+Nx-}\n\\begin{split}\n\\Big\\langle \\frac{N_x^+ N_x^-}{N_x^2} \\Big\\rangle_{N_x \\ge 1} &=\n\\sum_{N_x =1}^N \\binom{N}{N_x} \\Big(\\frac{2}{3} \\Big)^{N-N_x} \n\\sum_{N_x^+ =0}^{N_x} \\frac{N_x^+ N_x^-}{N_x^2} \\binom{N_x}{N_x^+}\n(q_x^+)^{N_x^+} (q_x^-)^{N_x^-}\\\\\n&=9q_x^+ q_x^- \\sum_{N_x =1}^N \\Big( 1-\\frac{1}{N_x} \\Big) \\binom{N}{N_x}\n\\Big(\\frac{2}{3} \\Big)^{N-N_x} \\Big(\\frac{1}{3} \\Big)^{N_x}\n=\\frac{1-x_0^2}{4} \\bigg( 1-\\Big(\\frac{2}{3}\\Big)^N -\\frac{F_N}{N} \\bigg),\n\\end{split}\n\\end{equation}\n\\end{widetext}\nwhere\n\\begin{equation}\\label{e:FN}\n\\begin{split}\nF_N &=N \\Big(\\frac{2}{3}\\Big)^N \\sum_{n =1}^{N} \\binom{N}{n} \\frac{1}{n} \n\\Big(\\frac{1}{2}\\Big)^{n} \\\\\n&=N \\Big(\\frac{2}{3}\\Big)^N \\bigg(\n\\sum_{n =1}^N \\frac{1}{n} \\Big(\\frac{3}{2}\\Big)^{n}\n-\\sum_{n =1}^{N} \\frac{1}{n} \\bigg).\n\\end{split}\n\\end{equation}\nHere the last equation was obtained by integrating the following sum with respect to $q$ from $0$ to $\\frac{1}{2}$:\n\\begin{equation}\n\\frac{d}{dq} \\sum_{n =1}^{N} \\binom{N}{n} \\frac{1}{n} q^{n}=\n\\frac{(1+q)^N -1}{q} =\n\\sum_{n=0}^{N-1} (1+q)^n.\n\\end{equation}\nUnfortunately the sum~\\eqref{e:FN} defining $F_N$ cannot be further simplified, but it can be shown that $\\lim_{N\\to \\infty} F_N =3$. Indeed, the second sum in \\eqref{e:FN} diverges only logarithmically, but it is multiplied with an exponentially descending factor, so it rapidly converges to zero. The first sum $S_N =N \\sum_{n=1}^N \\frac{1}{n} (2\/3)^{N-n}$ satisfies the recursion\n\\begin{equation}\nS_{N+1} -S_N = \\Big(\\frac{2}{3N} -\\frac{1}{3} \\Big)S_N +1,\n\\end{equation}\nwhich means that $\\lim_{N\\to \\infty} S_N =\\lim_{N\\to\\infty} F_N =3$.\n\nUsing~\\eqref{e:avNx+} and \\eqref{e:avNx+Nx-} the average of the error square of the $x$ component of the estimator is\n\\begin{equation}\n\\begin{split}\n\\langle \\Delta x^2 \\rangle &=\n\\Big\\langle \\frac{N_x^2 -4N_x^+ N_x^-}{N_x^2} \\Big\\rangle_{N_x \\ge 1} \n-\\langle x' \\rangle^2 \\\\\n&= \\frac{F_N}{N} (1-x_0^2) +x_0^2 \\Big(\\frac{2}{3} \\Big)^N \n\\bigg(1-\\Big(\\frac{2}{3}\\Big)^N \\bigg),\n\\end{split}\n\\end{equation}\nand similar formulas are valid in the other directions. By equation~\\eqref{e:d2} the variance of the estimator is:\n\\begin{equation}\\label{e:V6}\nV_{\\circledast} =\\frac{F_N (3-r_0^2)}{2N} +r_0^2 \\frac{6^N -4^N}{9^N} \\approx\n\\frac{9-3r_0^2}{2N}.\n\\end{equation}\nFor large $N\\gg 1$ this expression has the same asymptotic behavior as the variance~\\eqref{e:V3} of the estimator using projective spin measurements in orthogonal directions.\n\\subsection{Minimal qubit tomography \\label{ss:minqubt}}\nThis protocol was introduced in \\cite{MinQubit:04}. The protocol is based on a POVM, which is minimal in the sense that it contains only four positive operators $\\{Q_k\\}_{k=1}^4$ in the decomposition of unity. These operators are constant multiples of the four projectors $P_k =\\rho(\\mathbf{a}_k)$ being at the vertexes $\\{\\mathbf{a}_k\\}_{k=1}^4$ of a regular tetrahedron on the Bloch sphere, as shown in figure~\\ref{f:min}:\n\\begin{equation}\\label{e:a14}\n\\begin{aligned}\n\\mathbf{a}_1 &=\\frac{1}{\\sqrt{3}}\\begin{bmatrix} 1\\\\ 1\\\\ 1\\end{bmatrix},&\n\\mathbf{a}_2 &=\\frac{1}{\\sqrt{3}}\\begin{bmatrix} 1\\\\ -1\\\\ -1\\end{bmatrix},\n\\\\\n\\mathbf{a}_3 &=\\frac{1}{\\sqrt{3}}\\begin{bmatrix} -1\\\\ 1\\\\ -1\\end{bmatrix},&\n\\mathbf{a}_4 &=\\frac{1}{\\sqrt{3}}\\begin{bmatrix} -1\\\\ -1\\\\ 1\\end{bmatrix},\n\\end{aligned}\n\\end{equation}\n\\begin{align}\nQ_k &= \\frac{1}{4} \\rho(\\mathbf{a}_k) =\\frac{I+\\mathbf{a}_k \\bm{\\sigma}}{8},&\nI&= \\sum_{k=1}^4 Q_k.\n\\end{align}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.4]{mintom.eps}\n\\end{center}\n\\caption{\\label{f:min} The four projections $P_k$ at the vertexes $\\mathbf{a}_k$ of a regular tetrahedron on the Bloch sphere.}\n\\end{figure}\n\nDue to the tetrahedral symmetry, the vectors $\\mathbf{a}_k$ have the following properties:\n\\begin{align}\\label{e:propa}\n\\mathbf{a}_k \\mathbf{a}_l &= -\\frac{1}{3} +\\frac{4}{3} \\delta_{k,l}&\n\\sum_{k=1}^4 \\mathbf{a}_k \\circ \\mathbf{a}_k &=\\frac{4}{3} I.\n\\end{align}\n\nIn the state $\\rho_0 =\\rho(\\mathbf{r}_0)$ the probabilities $p_k$ of the four different measurement outcomes can easily be calculated with the help of the properties~\\eqref{e:sigma} of the Pauli matrices:\n\\begin{equation}\\label{e:p4}\np_k =\\Tr(\\rho_0 Q_k) =\\frac{\\Tr\\big((I+\\mathbf{r}_0 \\bm{\\sigma}) (I+\\mathbf{a}_k \\bm{\\sigma})\\big)}{8} =\n\\frac{1+\\mathbf{r}_0 \\mathbf{a}_k}{4}.\n\\end{equation}\nDenoting the number of $k$-th measurement results by $N_k$ and the total number of measurements by $N =\\sum_{k=1}^4 N_k$, the probability of a given measurement statistics in the state $\\rho_0$ is described again by the multinomial distribution:\n\\begin{equation}\n\\mathcal L(\\mathbf{r}_0) =\\frac{N!}{N_1! N_2! N_3! N_4!} \np_1^{N_1} p_2^{N_2} p_3^{N_3} p_4^{N_4}.\n\\end{equation}\nThis likelihood function takes its maximal value at the zero of its gradient, which yields the equation\n\\begin{equation}\\label{e:lemin}\n\\sum_{k=1}^{4} \\frac{N_k \\mathbf{a}_k}{1+\\mathbf{r}' \\mathbf{a}_k} =0\n\\end{equation}\nfor the Bloch vector $\\mathbf{r}'$ of the estimator $\\rho' =\\rho(\\mathbf{r}')$. For general $\\mathbf{a}_k$ vectors it is hopeless to solve explicitly this equation, but for the symmetrically distributed vectors~\\eqref{e:a14}, using their properties~\\eqref{e:propa} it is easy to check that the unique solution is \n\\begin{equation}\\label{e:rv4}\n\\mathbf{r}' =3\\sum_{k=1}^{4} \\frac{N_k}{N} \\mathbf{a}_k.\n\\end{equation}\nThis solution is well defined for any measurement statistics $\\{ N_k\\}$, but it may fall out of the Bloch ball.\n\nUsing the multinomial theorem it is easy to show that $\\langle N_k \\rangle =Np_k$, so\n\\begin{equation}\n\\langle \\mathbf{r}' \\rangle =3\\sum_{k=1}^4 p_k \\mathbf{a}_k =\n\\frac{3}{4} \\sum_{k=1}^4 \\mathbf{a}_k + \n\\frac{3}{4}\\bigg(\\sum_{k=1}^4 \\mathbf{a}_k \\circ \\mathbf{a}_k \\bigg) \\mathbf{r}_0 =\\mathbf{r}_0,\n\\end{equation}\nwhich means that the estimator $\\rho'$ is unbiased.\n\nApplying again the multinomial theorem, the formulas~\\eqref{e:p4} for the probabilities and the properties~\\eqref{e:propa} of the $\\mathbf{a}_k$ vectors, we obtain that\n\\begin{equation}\n\\begin{split}\n\\Big\\langle \\sum_{k=1}^4 N_k^2 \\Big\\rangle &=\n\\sum_{k=1}^4 \\Big( \\big\\langle N_k (N_k -1)\\big\\rangle +\\langle N_k \\rangle \\Big)\\\\\n&=\\sum_{k=1}^4 \\big( N(N-1)p_k^2 + N p_k \\big)\\\\\n&=\\frac{3+r_0^2}{12}(N^2 -N) +N.\n\\end{split}\n\\end{equation}\nFrom this, using equations~\\eqref{e:rv4} and \\eqref{e:propa} we immediately get that\n\\begin{equation}\n\\big\\langle r'^2 \\big\\rangle =\n\\frac{12}{N^2} \\Big\\langle \\sum_{k=1}^4 N_k^2 \\Big\\rangle-3 =\nr_0^2 +\\frac{9-r_0^2}{N},\n\\end{equation}\nand finally, using~\\eqref{e:d2}, for the variance we obtain:\n\\begin{equation}\\label{e:V4}\nV_{\\otimes} =\\frac{\\big\\langle r'^2 \\big\\rangle -\\langle \\mathbf{r}' \\rangle^2}{2} =\n\\frac{9-r_0^2}{2N}.\n\\end{equation}\n\nComparing it with the variances~\\eqref{e:V3} and \\eqref{e:V6} obtained for the previous protocols, we see that in all cases the variances decrease as $1\/N$, but for states close to pure states (i.e., if $r_0 \\lessapprox 1$) the minimal qubit tomography protocol is less performant by a factor of $\\frac{9-1}{9-3}= \\frac{4}{3}$.\n\n\\section{Continuous, equivariant measurement \\label{s:contmeas}}\n\nUp to this point we have investigated three different qubit tomography protocols, which were based on (either projective or POVM) measurements with only a finite number of possible outcomes. In a finite $d$ dimensional Hilbert space the maximal number of possible outcomes of a projective measurement is $d$, but this restriction does not hold for generalized, POVM measurements. In this section we construct a new qubit tomography protocol based on a \\emph{continuous} POVM measurement, which is \\emph{equivariant} under the symmetry group of the qubit system, and supported by the set of pure states, i.e., by the Bloch sphere.\n\nThe attribute ``continuous'' refers to the fact that the investigated measurement has an infinite number of possible outcomes, which form a topological manifold---the Bloch sphere in our case. The attribute ``equivariant'' refers to a kind of nice symmetry property, which deserves a deeper investigation.\n\n\\subsection{Equivariance \\label{ss:equiv}}\n\nAt an abstract level the symmetry group $G$ of an (either classical or quantum) system is an automorphism group of its event lattice \\cite{BirkNeu:36,JvN:55}. States are probability measures on the event lattice, so $G$ acts both on events and (by pullback construction) on states. In quantum mechanics events are projections in the Hilbert space $\\mathcal H$ of the system, and by Gleason's theorem \\cite{Gleas:57} states are represented by density operators. By Wigner's theorem~\\cite{Wig:31}, if $\\dim \\mathcal H >2$, every automorphism $g$ of the projector lattice can be realized by conjugation with a unitary (or antiunitary) operator $U_g$, thus we obtain a left $G$-action on the lattice of projections (events) $\\mathcal P$, and a right $G$-action on the set of density operators (states) $\\mathcal S$:\n\\begin{align}\\label{e:GactP}\nG\\times \\mathcal P &\\to \\mathcal P.&\n(g,P) &\\mapsto gP=U_g PU_g^*;\n\\\\ \\label{e:GactS}\n\\mathcal S \\times G &\\to \\mathcal S,&\n(\\rho,g) &\\mapsto \\rho g=U_g^* \\rho U_g,\n\\end{align}\nand, by the pullback construction\n\\begin{equation}\\label{e:rhogp}\n\\rho(gP) =\\Tr(\\rho U_g P U_g^* )= \\Tr(U_g^* \\rho U_g P) =(\\rho g)(P).\n\\end{equation}\n\nPerforming many times a measurement with possible outcomes in the Borel space $(S,\\mathcal B)$ we obtain a probability distribution on the space $S$. Thus a measurement can be regarded as assigning to a quantum state $\\rho$ (i.e., probability function on $\\mathcal P$) a classical state $p_{\\rho}$ (i.e., a probability function on $\\mathcal B$). The simplest way to do this is pulling back $\\rho$ by a morphism $P:\\mathcal B \\to \\mathcal P$ between the classical and quantum event lattice:\n\\begin{align}\np_{\\rho} =\\rho \\circ P: \\mathcal B &\\to [0,1], &\nE &\\mapsto p_{\\rho}(E) =\\Tr\\big(\\rho P(E)\\big).\n\\end{align}\nThe classical $\\to$ quantum lattice morphism $P$ is nothing but a projector valued measure, and the above formula is exactly the probability for the outcome of the projective measurement $P$ being in $E$. Formally the generalized, POVM measurements are obtained by changing $P$ to a positive operator valued measure $Q$ on $(S,\\mathcal B)$.\n\nNow let us assume that $S$ is a right $G$-space, and let $\\gamma_g :S \\to S$ denote the transformation corresponding to $g\\in G$. (So $\\gamma_g \\circ \\gamma_h =\\gamma_{hg}$.) Then its inverse image $\\gamma_g^{-1}: \\mathcal B \\to \\mathcal B$ defines a left action of $G$ on $\\mathcal B$. We say that the POVM $Q$ is \\emph{equivariant} with respect to the group actions $\\gamma^{-1}$ and $\\Hat{U} =U\\cdot U^*$ (conjugation by $U$) if for any $E\\in \\mathcal B$ and $g\\in G$ the following diagram commutes:\n\\begin{align}\\label{e:genimp}\n\\begin{CD}\n\\mathcal B @>{\\gamma_g^{-1}}>> \\mathcal B\\\\\n@VQVV @VQVV \\\\\nB(\\mathcal H) @>{\\Hat{U}_g}>> B(\\mathcal H)\n\\end{CD}\n&&\nU_g Q(E) U_g^* &= Q\\big( \\gamma_g^{-1}(E) \\big),\n\\end{align}\nwhere $B(\\mathcal H)$ denotes the set of bounded operators in $\\mathcal H$.\n\nFor projector valued measures this structure was discovered by George W. Mackey \\cite{Mac:76,Mac:78}, while he was studying the foundations of quantum mechanics \\cite{Mac:63}, and he called it \\emph{imprimitivity system}. Mackey, in his \\emph{imprimitivity theorem}, classified and explicitly constructed the possible imprimitivity systems using \\emph{induced representations}. Since then induced representation became a very important tool in the theory of unitary group representation and harmonic analysis, while the imprimitivity theorem turned out to be a cornerstone of quantum mechanics of free systems. Indeed, the equivariance condition~\\eqref{e:genimp}, stated for projector valued measures on the three dimensional Euclidean space, is a very general and deep expression of the canonical commutation relations. Starting from this, many important properties (like the existence of mass, spin) of the elementary particles can be deduced, based on purely spacetime symmetry requirements \\cite{Mac:63,Mac:78,Varad:68,Jau:68}.\n\nBorrowing the terminology from the projective case, we refer to the equivariance condition~\\eqref{e:genimp} as \\emph{general system of imprimitivity}, more precisely, as the POVM $Q$ is a \\emph{generalized system of imprimitivity for $U$ based on $S$}. In the next subsection we give a very simple example for this in terms of a qubit.\n\n\\subsection{Equivariant POVM on the Bloch sphere \\label{ss:equPOVM}}\n\nFirst of all, we note that Wigner's theorem does not apply for two dimensional Hilbert spaces; the projector lattice of $\\mathbb{C}^2$ has a much bigger symmetry group than the symmetries induced by unitaries (and antiunitaries). People consider it as a pathological fact due to the low dimensionality of the space. Here we join this opinion and consider the group $SO(3,\\mathbb{R}) =SU(2,\\mathbb{C})\/\\mathbb{Z}_2$ as the symmetry group $G$ of the qubit. (The reason for the factorization is that conjugation with $U$ and $-U$ is the same transformation.)\n\nFurthermore, as the set $S$ of possible measurement outcomes we choose the set of pure states, i.e., the Bloch sphere itself! It is a very clever choice for several reasons: $i)$~By~\\eqref{e:GactS} $S$ is a right $G$-space in a natural way. $ii)$~Classically pure states are Dirac measures on the phase space, thus in this respect $S$ is the quantum analog of classical phase space. Is there any better measurement than the one returning \\emph{the} (or \\emph{a}?) phase space position (pure state) of the measured system? And finally $iii)$~pure states are rank one projections, i.e., positive operators, thus the POVM $Q$ is ``already there'' on $S$, one has only to normalize it.\n\nThe rank one projectors $\\rho(\\mathbf{s})$ are parametrized by points $\\mathbf{s}$ of the Bloch sphere ($|\\mathbf{s}|=1$), unitaries have the form $U_{\\mathbf{n}} =e^{i\\mathbf{n}\\bm{\\sigma}}$ (where $|\\mathbf{n}| < 2\\pi$), and it can be shown that\n\\begin{equation}\\label{e:UrhoU}\nU_{\\mathbf{n}} \\rho(\\mathbf{s}) U_{\\mathbf{n}}^* =\ne^{i\\mathbf{n}\\bm{\\sigma}} \\rho(\\mathbf{s}) e^{-i\\mathbf{n}\\bm{\\sigma}}=\n\\rho(O_{-2\\mathbf{n}} \\mathbf{s}),\n\\end{equation}\nwhere $O_{\\mathbf{m}}$ is the orthogonal rotation around the axis $\\mathbf{m}$ at an angle $|\\mathbf{m}|$. (To derive this the equation $[\\mathbf{n}\\bm{\\sigma}, \\mathbf{m} \\bm{\\sigma}] =2i(\\mathbf{n} \\times \\mathbf{m}) \\bm{\\sigma}$ is needed, which is a straightforward consequence of~\\eqref{e:sigma}.) Thus in the Bloch sphere picture the symmetry transformations are the rotations of the sphere! (We remark here that the nonunitary symmetries are arbitrary homeomorphisms of the Bloch sphere which map antipodal points to antipodal points.) Equations~\\eqref{e:GactS} and \\eqref{e:UrhoU} also show that in this case the right $G$-action $\\gamma_{\\mathbf{n}}$ on the space $S$ is\n\\begin{equation}\\label{e:gam}\n\\gamma_{\\mathbf{n}}(\\mathbf{s}) =\\text{representing point of }U_{\\mathbf{n}}^* \\rho(\\mathbf{s}) U_{\\mathbf{n}} =O_{2\\mathbf{n}} \\mathbf{s}.\n\\end{equation}\n\nNow let $\\omega$ be the normalized area measure on the unit sphere $S$, i.e., in spherical polar coordinates $(\\vartheta, \\varphi)$, $d\\omega(\\vartheta, \\varphi) =\\frac{\\sin \\vartheta}{4\\pi} d\\vartheta d\\varphi$, and for a Borel set $E\\subset S$ let the POVM $Q$ be defined by\n\\begin{subequations}\\label{e:Q}\n\\begin{equation}\\label{e:Qa}\nQ(E) =2\\int_{\\mathbf{s}\\in E} \\rho(\\mathbf{s}) d\\omega(\\mathbf{s}) \n=2\\omega(E) \\rho\\big(\n{\\textstyle \\int_{\\mathbf{s} \\in E} \\mathbf{s} d\\omega(\\mathbf{s})}\\big),\n\\end{equation}\nor shortly\n\\begin{align}\\label{e:Qb}\ndQ(\\mathbf{s}) &=2\\rho(\\mathbf{s}) d\\omega(\\mathbf{s}),&\nQ &= 2\\rho \\omega. \n\\end{align}\n\\end{subequations}\n(The last equality in~\\eqref{e:Qa} is a consequence of~\\eqref{e:rhoav}.) It is clear that $Q$ is normalized, i.e., $Q(S) =2\\rho(\\mathbf{0}) =I$.\n\nUsing the previous three formulas~(\\ref{e:UrhoU}--\\ref{e:Q}) and the invariance of the area measure $\\omega =\\omega\\circ O^{-1}$ under any rotation $O$, it is simple to show that $Q$ satisfies the equivariance condition~\\eqref{e:genimp}:\n\\begin{multline}\nU_{\\mathbf{n}} Q(E) U_{\\mathbf{n}}^* =\n2{\\textstyle \\omega(E)\\rho\\big( \\int_{\\mathbf{s} \\in E} O_{-2\\mathbf{n}} \\mathbf{s} d\\omega(\\mathbf{s}) \\big)}\n\\\\\n=2{\\textstyle \\omega(E)\\rho\\big( \\int_{\\mathbf{s} \\in O_{-2\\mathbf{n}}E} \\mathbf{s} d\\omega(\\mathbf{s}) \\big)}\n=Q\\big( \\gamma_{\\mathbf{n}}^{-1}(E)\\big).\n\\end{multline}\n(It is worth noticing that here essentially the equation~\\eqref{e:rhogp} and the invariance of $\\omega$ was used.)\n\nWe give the concrete formula of the POVM $Q$ in spherical polar coordinates $(\\vartheta, \\varphi)$, although we will keep on using the abstract form~\\eqref{e:Q}:\n\\begin{equation}\ndQ(\\vartheta, \\varphi) =\\frac{\\sin\\vartheta}{4\\pi}\n\\begin{bmatrix}\n1+\\cos\\vartheta & e^{-i\\varphi} \\sin\\vartheta\\\\\ne^{i\\varphi} \\sin\\vartheta & 1-\\cos\\vartheta\n\\end{bmatrix}\n d\\vartheta d\\varphi.\n\\end{equation}\n\nIntuitively this POVM can be regarded as the limit of discrete POVM's supported by more and more points scattered uniformly on the Bloch sphere. In the previous section we have seen examples for four (subsection~\\ref{ss:minqubt}) and six (subsection~\\ref{ss:hexqubt}) supporting points with tetrahedral and hexagonal symmetry, respectively.\n\nNow let us investigate the distribution of the measurement $Q$ provided that the system is in state $\\rho_0 =\\rho(\\mathbf{r}_0)$. The probability density function $f_0$ (with respect to $\\omega$) at the outcome $\\mathbf{s} \\in S$ is\n\\begin{equation}\\label{e:f0}\n\\begin{split}\nf_0 (\\mathbf{s}) &=\\lim_{dE\\to 0}\\frac{\\Tr\\big(\\rho_0 Q(dE)\\big)}{\\omega(dE)}\\\\\n&=2\\Tr\\big(\\rho(\\mathbf{r}_0)\\rho(\\mathbf{s})\\big) \n= 1+\\mathbf{r}_0 \\mathbf{s},\n\\end{split}\n\\end{equation}\nwhere $dE$ is an infinitesimally small area on the sphere $S$ around $\\mathbf{s}$. (The last equality is a direct consequence of~\\eqref{e:sigma}.) This probability distribution is \\emph{unsharp}, even for pure states! If $r_0 =1$, then the maximum is $f_0(\\mathbf{r}_0) =2$, and the minimum is $f_0(-\\mathbf{r}_0) =0$.\n\nHere it is worth noting that the POVM $Q$ defined in~\\eqref{e:Q} is not the unique solution of the equivariance condition \\eqref{e:genimp} (but probably the most reasonable one). Indeed, for any $\\alpha \\in [0,1]$ the POVM\n\\begin{equation}\nQ_{\\alpha} (E) =\\alpha Q(E) + (1-\\alpha)\\omega(E) I\n\\end{equation}\nsatisfies the equivariance condition~\\eqref{e:genimp}, but the density function of $\\Tr\\big(\\rho_0 Q_{\\alpha}(\\cdot )\\big)$ at $\\mathbf{s} \\in S$ is $1+\\alpha \\mathbf{r}_0 \\mathbf{s}$, which is even less sharp then~\\eqref{e:f0}.\n\nFinally it is worth calculating the measure of a semisphere. The ``center of mass'' of the  semisphere $S^+ (\\mathbf{r})$ with midpoint $\\mathbf{r}$ ($|\\mathbf{r}|=1$) is at $\\frac{\\mathbf{r}}{2}$, so by~\\eqref{e:Qa}\n\\begin{equation}\nQ\\big( (S^+ (\\mathbf{r})\\big) =\\rho\\Big( \\frac{\\mathbf{r}}{4}\\Big) =\n\\frac{I}{2}+ \\frac{\\mathbf{r} \\bm{\\sigma}}{4}.\n\\end{equation}\nComparing it to the orthogonal projections~\\eqref{e:Pxyzpm}, we see that if we are interested in a particular spin component of the state $\\rho_0$, then it is better to perform a projective spin measurement than measuring $Q$. But what if we are interested in all components of the vector $\\mathbf{r}_0$?\n\n\\subsection{Maximal qubit tomography}\n\nIn this subsection we investigate a quantum bit state estimation protocol based on the POVM $Q:\\mathcal B(S)\\to B(\\mathbb{C}^2)$ introduced in the previous subsection [see equation~\\eqref{e:Q}]. We find it convenient to call this protocol \\emph{maximal qubit tomography}, since the whole set $S$ of pure states constitute the possible measurement outcomes.\n\nAssume that performing $N$ independent measurements on replicas of the qubit in the same state $\\rho_0$, the measurement outcomes $\\{ \\mathbf{n}_k \\}_{k=1}^N \\subset S$ are obtained, where each $\\mathbf{n}_k$ represents a pure state on the Bloch sphere $S$, i.e., $\\mathbf{n}_k \\in \\mathbb{R}^3$, $|\\mathbf{n}_k|=1$. Using \\eqref{e:f0}, the likelihood function, i.e., the probability density function on $S^N$ of obtaining this measurement statistics in the state $\\rho_0 =\\rho(\\mathbf{r}_0)$ is\n\\begin{equation}\\label{e:Lmax}\n\\mathcal L(\\mathbf{r}_0) = \\prod_{k=1}^N (1+\\mathbf{r}_0 \\mathbf{n}_k ).\n\\end{equation}\nIts gradient at $\\mathbf{r}'$ is $\\mathcal L(\\mathbf{r}')\\sum_{k=1}^N \\frac{\\mathbf{n}_k}{1+\\mathbf{r}' \\mathbf{n}_k}$, which yields the likelihood equation\n\\begin{equation}\\label{e:lemax}\n\\sum_{k=1}^N \\frac{\\mathbf{n}_k}{1+\\mathbf{r}' \\mathbf{n}_k}=0.\n\\end{equation}\n\nThe second derivative of the logarithm of the likelihood function~\\eqref{e:Lmax}\n\\begin{equation}\n\\frac{\\partial^2}{\\partial \\mathbf{r}^2} \\ln\\mathcal L(\\mathbf{r}) =\n- \\sum_{k=1}^N \\frac{\\mathbf{n}_k \\circ \\mathbf{n}_k}{(1+\\mathbf{r} \\mathbf{n}_k)^2}\n\\end{equation}\nis everywhere negative definite, so the likelihood equation~\\eqref{e:lemax} \\emph{has a unique} solution for $\\mathbf{r}'$, where $\\mathcal L(\\mathbf{r}')$ is maximal. Unfortunately for a general measurement statistics $\\{\\mathbf{n}_k\\}_{k=1}^N$ this solution cannot be analytically determined. (In subsection~\\ref{ss:minqubt} a similar equation~\\eqref{e:lemin} was obtained, which could be explicitly solved \\eqref{e:rv4} because of the symmetry of the fixed $\\{\\mathbf{a}_k\\}_{k=1}^4$ vectors.)\n\nInstead of using the maximum likelihood estimator we introduce another obvious estimator\n\\begin{align}\\label{e:rpmax}\n\\mathbf{r}' &=f(N) \\sum_{k=1}^N \\mathbf{n}_k,&\n&\\text{with}&\nf(N) &= \\frac{3}{N}\n\\end{align}\nwhere the (yet) unknown coefficient $f(N)$ is determined from the expectational value of $\\mathbf{r}'$. For this aim we need two simple integrals:\n\\begin{align}\\label{e:ints}\n\\int_{\\mathbf{n} \\in S} \\mathbf{n} d\\omega(\\mathbf{n}) &=0,&\n\\int_{\\mathbf{n} \\in S} \\mathbf{n} \\circ \\mathbf{n} d\\omega(\\mathbf{n}) &= \\frac{1}{3} I,\n\\end{align}\nwhere $\\omega$ is the normalized area on the unit sphere $S$. (The first integral is zero by symmetry, and the second integral can easily be calculated in spherical polar coordinates $(\\vartheta,\\varphi)$, where $d\\omega(\\vartheta,\\varphi) =\\frac{\\sin \\vartheta}{4\\pi} d\\vartheta d\\varphi$.)\n\nThus, using \\eqref{e:Lmax}, \\eqref{e:rpmax}, and then \\eqref{e:ints}, the expectational value of the estimator is\n\\begin{equation}\n\\begin{split}\n\\langle \\mathbf{r}' \\rangle &=\nf(N) \\sum_{k=1}^N \\int_{\\{\\mathbf{n}_k\\} \\in S^N}\n\\mathcal L(\\mathbf{r}_0) \\mathbf{n}_k d^N\\omega(\\mathbf{n}_k) \\\\\n&= Nf(N) \\int_{\\mathbf{n} \\in S} \\mathbf{n} (1+\\mathbf{r}_0 \\mathbf{n}) d\\omega(\\mathbf{n}) =\n\\frac{Nf(N)}{3} \\mathbf{r}_0,\n\\end{split}\n\\end{equation}\nwhich means that indeed, $f(N)=\\frac{3}{N}$ should be chosen to get an unbiased estimation. (The integrals decouple, and for every $k$ only the integral over $\\mathbf{n}_k$ gives a nontrivial result, the other integrals are $1$. The obtained result is also in accordance with~\\eqref{e:rv4}.)\n\nIn order to calculate the variance we need another integral, which can be easily obtained from~\\eqref{e:ints}:\n\\begin{equation}\n\\int\\limits_{\\mathbf{n} \\in S} \\int\\limits_{\\mathbf{m} \\in S}\n\\mathbf{n}\\mathbf{m} (1+\\mathbf{r}_0 \\mathbf{n}) (1+\\mathbf{r}_0\\mathbf{m} ) d\\omega(\\mathbf{m}) d\\omega(\\mathbf{n})\n=\\frac{r_0^2}{9}.\n\\end{equation}\nUsing this, we get that\n\\begin{equation}\n\\begin{split}\n\\big\\langle r'^2 \\big\\rangle &=\n\\frac{9}{N^2} \\int_{\\{\\mathbf{n}_k\\} \\in S^N} \n\\Big(\\sum_{k=1}^N \\mathbf{n}_k \\Big)^2 \\mathcal L(\\mathbf{r}_0) d^N\\omega(\\mathbf{n}_k) \\\\\n&=\\frac{9 +(N-1)r_0^2}{N},\n\\end{split}\n\\end{equation}\nso, by~\\eqref{e:d2}, the variance of the (unrestricted) estimator~\\eqref{e:rpmax} is\n\\begin{equation}\\label{e:Vmax}\nV_{\\Circle} =\\frac{\\big\\langle r'^2 \\big\\rangle -\\langle \\mathbf{r}'\\rangle^2}{2} \n= \\frac{9-r_0^2}{2N},\n\\end{equation}\nwhich exactly coincides with the one~\\eqref{e:V4} obtained for the minimal qubit tomography protocol.\n\n\\section{Conclusions}\n\nIn table \\ref{t:res} we summarize the results obtained for the four different qubit state estimation protocols. (The variance of the state $\\rho(\\mathbf{r}')$ is calculated from the Hilbert-Schmidt distance, and by equation~\\eqref{e:d2} it is half of the variance of $\\mathbf{r}'$.)\n\n\\begin{table*}\n\\begin{tabular}{|ll|c|c|}\n\\hline\n&\n\\parbox{9cm}{\\vspace{2pt}\\textbf{Protocol}\\vspace{2pt}} & \n\\textbf{mean} &\n\\textbf{\\boldmath variance of $\\rho'$}\n\\\\ \n\\hhline{|==|=|=|}\n$\\ast$ &\n\\parbox{9cm}{projective spin measurements in three orthogonal directions} &\n\\parbox{2.2cm}{unbiased}& \n\\parbox{4cm}{\\vspace{2pt}\n$\\displaystyle \\frac{9-3r_0^2}{2N}$\n\\vspace{2pt}}\n\\\\ \\hline\n$\\circledast$ &\n\\parbox{9cm}{POVM measurement in six orthogonal directions} &\n\\parbox{2.2cm}{asymptotically unbiased} &\n\\parbox{4cm}{\\vspace{2pt}\n$\\displaystyle \\frac{9-3r_0^2}{2N} \\quad\\text{for}\\quad N\\gg 1$\n\\vspace{2pt}}\n\\\\ \\hline\n$\\otimes$ &\n\\parbox{9cm}{\\emph{minimal qubit tomography}\\\\\n(POVM measurement in four tetrahedral directions)} &\n\\parbox{2.2cm}{unbiased} &\n\\parbox{4cm}{\\vspace{2pt}\n$\\displaystyle \\frac{9-r_0^2}{2N}$\n\\vspace{2pt}}\n\\\\ \\hline\n$\\Circle$ &\n\\parbox{9cm}{\\emph{maximal qubit tomography}\\\\\n(POVM measurement uniformly on the whole Bloch sphere)} &\n\\parbox{2.2cm}{unbiased} &\n\\parbox{4cm}{\\vspace{2pt}\n$\\displaystyle \\frac{9-r_0^2}{2N}$\n\\vspace{2pt}}\n\\\\ \\hline\n\\end{tabular}\n\\caption{\\label{t:res}The results obtained for the different qubit tomography protocols.}\n\\end{table*}\n\nIt is clearly visible that all estimation schemes perform more or less equally well, although there are projective ($\\ast$), discrete POVM ($\\circledast$, $\\otimes$), and continuous POVM measurements ($\\Circle$) among them. For states close to pure states (i.e., for $r_0 \\lessapprox 1$) the variance of the first two protocols with hexagonal symmetry ($\\ast$ and $\\circledast$) is a bit less than the variance of $\\otimes$ and $\\Circle$. On the other hand, the variance of the minimal and maximal qubit tomography protocol is exactly the same for all states.\n\nBased upon these examples we may draw some general conclusions on the unusual properties of POVM measurements.\n\nIn classical Hamiltonian mechanics the phase space can be identified with the set of pure states of the system, which are simply Dirac measures concentrated on a single point of the phase space. This observation motivates to regard the set of pure states even in a finite dimensional quantum mechanical system as the quantum analog of phase space.\n\nA widespread paradigm of quantum mechanics, based on Heisenberg's uncertainty principle, is the fact that all the classical phase space variables cannot be measured simultaneously with arbitrary high precision. Originally this was stated for \\emph{projective measurements}, which are \\emph{sharp} in the sense that for every possible measurement outcome there \\emph{is} a state for which the specified outcome occurs with probability one. (Right after the measurement the system ``jumps'' into a state of this kind.) Coordinate and momentum operators do not commute, thus they do not possess a common projector valued measure.\n\nOn the other hand, there are generalized POVM measurements which yield results corresponding to simultaneous values of noncommuting operators! This fact is, however, not in contradiction with Heisenberg's uncertainty principle, since the result of the POVM measurement is \\emph{unsharp}, i.e., for \\emph{any} pure state of the system the measurement results have dispersed probability distributions. A very simple example is the POVM measurement constructed in subsection~\\ref{ss:equPOVM}, which clearly demonstrates all the unusual features of generalized measurements. In summary, in contrast to projective measurements, a POVM measurement\n\\begin{itemize}\n\\item[$\\ddot\\smile$]\nmay have a continuous set of possible outcomes, even in a finite dimensional Hilbert space;\n\\item[$\\ddot\\smile$]\nmay yield simultaneously values for incompatible (noncommuting) observables;\n\\item[$\\ddot\\frown$]\nbut these simultaneous results are unsharp. (In particular, it means that repeated measurements do not give the same outcome.)\n\\item[$\\ddot-$]\nFurthermore, the information gains ($\\ddot\\smile$) and information losses ($\\ddot\\frown$) somehow compensate each other, so cleverly chosen POVM measurements yield more or less the same amount of information about the true state as projective measurements.\n\\item[$\\ddot\\smile$]\nLast but not least, POVM measurements can be much better adjusted to respect continuous symmetries of the system than projective measurements.\n\\end{itemize}\n\nFinally we remark that POVM measurements (or \\emph{weak measurements}) are not pure mathematical constructions, they can also be experimentally realized by making an appropriate projective measurement on a composite system.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe Roofline performance model \\cite{CACM09_Roofline1} offers an insightful and intuitive way to extracte key computational characteristics for applications in high-performance computing (HPC).\nIts capability to abstract away the complexity of modern memory hierarchies and guide performance analysis and optimization effort has gained its popularity in recent years.\n\n\nRoofline is a throughput-oriented model centered around the interplay between computational capabilities, memory bandwidth, and data locality.  Data locality is the reuse of data once it is being loaded from the main memory, and it is commonly expressed as the arithmetic intensity (AI), ratio between the floating-point operations performed and the data moved (FLOPs\/Byte).\nThe performance (GFLOP\/s) is bound by the following two terms:\n\\begin{equation}\n\\mbox{GFLOP\/s} \\leq \\mbox{min} \\left\\{ \n  \\begin{array}{@{}l@{}}\n    \\mbox{Peak GFLOP\/s}\t\t\t\t\t\t\t\\\\\n    \\mbox{Peak GB\/s} \\times \\mbox{Arithmetic Intensity}\t\n  \\end{array}\n\\right.\n\\label{eq:roofline_naive}\n\\end{equation}\n\n\n\n\n\nConventionally, the Roofline model is focused on one level of the memory system, but this has been extended to the entire memory hierarchy in recent years, named the hierarchical Roofline model. \nThe hierarchical Roofline helps understand cache reuse and data locality and provides additional insights into the efficiency of the application's utilization of the memory subsystem. \nThe hierarchical Roofline has been integrated into Intel Advisor \\cite{advisor,koskela2018novel}, and NVIDIA Nsight Compute \\cite{ncu,nsight_compute}. \nEven though they should be the go-to methods for Roofline analysis, we would like to present in this paper a few other tools for the purpose of flexibility and generality. \n\nWe will discuss the use of Intel Advisor \\cite{advisor}, RRZE LIKWID \\cite{likwid}, Intel SDE \\cite{SDE} and Intel VTune \\cite{VTune} on Intel CPUs, and nvprof \\cite{nvprof}, Nsight Compute metrics, and Nsight Compute section files \\cite{nsight_compute} on NVIDIA GPUs.\nA mini-application will be used for demonstration and validation purpose, and it is extracted from the Material Science code BerkeleyGW \\cite{BGW2} called General Plasmon Pole (GPP) \\cite{examplescripts}. \nArchitecture-wise, we will focus on the Intel Knights Landing (KNL) CPU and NVIDIA V100 GPU.  \n\n \n\n  \n\nTo facilitate the Roofline study, a range of other tools have sprung to life as well,\nfor example, the Empirical Roofline Toolkit (ERT) for more accurate machine characterization \\cite{ert,yang2018empirical}, and \\cite{nerscroofline,yang2018toward,madsen2020timemory,yang2019hierarchical,yang2019cug} for more streamlined data collection methods.\nOther than tools development, there are many studies on the application of the Roofline model in traditional HPC \\cite{Doerfler,yang2019hierarchical,yang2019cug,del2020accelerating,gayatri2018case} and Machine Learning \\cite{yang2019hierarchical,yang2019cug,wang2020pmbs,wang2020dlonsc}, and extension and refinement of the model to related topics in HPC, such as instruction Roofline \\cite{ding2019instruction}, time-based Roofline \\cite{wang2020dlonsc}, Roofline scaling trajectory \\cite{ibrahim2019performance}, performance portability analysis based on Roofline \\cite{yang2018empirical}, and power and energy Roofline  \\cite{powerroofline,alexpowerroofline}.\n\n\n\n\\section{Application and Machine Setup}\n\n\\subsection{Mini-Application General Plasmon Pole (GPP) \\label{sec:gpp}}\n\nThe GPP mini-application~\\cite{examplescripts} is extracted from the Material Science code BerkeleyGW~\\cite{BGW2}, and it represents the work typically done on a single MPI rank.\nIt is written in C++, and parallelized with OpenMP on the CPU and CUDA on the GPU.\nThe computation involved this mini-app is tensor-contraction like, and several pre-calculated complex double precision arrays are multiplied and summed over certain dimensions and collapsed into a small vector. \nThe problem used in this paper is a medium sized one, and it comprises of 512 electrons and 32768 plane wave basis elements.\nThe pseudo code for this mini-app is as follows.\n\n{\\footnotesize\n\\begin{verbatim}\n  do band = 1, nbands   \n   do igp = 1, ngpown  \n    do ig = 1, ncouls   \n     do iw = 1, nw     \n       load wtilde_array(ig,igp)  \n       load aqsntemp(ig,band)  \n       load eps(ig,igp)  \n       compute wdiff, delw, sch_array, ssx_array\n       reduce on achtemp(iw), asxtemp(iw)\n\\end{verbatim}\n}\nThe real code, job scripts and resulted are available at \\cite{examplescripts}.\n\n \n\\subsection{Machine Setup\\label{sec:machine}}\n\nThis study is conducted on the Cori supercomputer at the National Energy Research Scientific Computing Center (NERSC) at Lawrence Berkeley National Laboratory (LBNL).\n\nCori has three main partitions, Haswell, KNL and GPU, and this study has used its KNL partition \\cite{nerscdocu1} and GPU chassis \\cite{nerscdocu2}. \nEach KNL node is a single-socket Intel Xeon Phi Processor 7250 (Knights Landing) processor and has 68 physical cores.\nThere is 96 GB DDR4 memory and 16 GB MCDRAM (or HBM) per node, with the MCDRAM configured in `cache' mode by default.\nThe GPU chassis is deployed primarily for \nthe NERSC Exascale Science Applications Program (NESAP).\nIt has 18 GPU nodes in total, and each node contains two 20-core Intel Xeon Gold 6148 Skylake CPUs, 384 GB DDR4 memory, \nand 8 NVIDIA V100 Volta GPUs.\nEach GPU has 80 Streaming Multiprocessors (SMs), 16 GB HBM2 memory, and is connected to others in a `hybrid cube-mesh' topology.\n\n\\section{Methods and Results}\n\n\\subsection{Roofline Data Collection on Intel CPUs\\label{sec:metho1}}\n\nIntel Advisor \\cite{advisor} provides the production quality, fully integrated hierarchical Roofline analysis on Intel CPUs, with very little user effort required. Compared to LIKWID \\cite{likwid}, it has a higher profiling overhead due to the static instruction analysis and cache simulation. LIKWID \\cite{likwid} is an open-source package developed at the Regional Computing Center Erlangen (RRZE) in Germany. It provides several \n`performance groups' for easier and more streamlined performance analysis, and in this paper, we have identified a few for the hierarchical Roolfine data collection.\nLIKWID uses metrics that are based on micro-ops not instructions, and in some cases, it does not distinguish the different vector lengths, such as scalars, AVX-2\/AVX-512 instructions, and masked\/unmasked vector lanes.\nThis may cause certain inaccuracy and require extra care, however its low overhead has made it a very attractive option for large-scale application analysis.\nTo collect hierarchical Roofline data, another method is to use Intel SDE \\cite{SDE} and VTune \\cite{VTune}.\nSDE has a very high profiling overhead but it provides the most accurate instruction count and it can produce L1 data movement information as well. \nOn the other hand, VTune can be used to collect DDR\/MCDRAM information to complement SDE.\nIn the following few subsections, we will detail the command lines used to collect Roofline data on KNL and the subsequent results.\n\n\\subsubsection{Intel Advisor}\n\nAdvisor can be invoked as follows for Roofline analysis, and Fig.~\\ref{fig:adv} shows that in GPP, the most significant function takes 2s of `Self-Time' and produces 398 GFLOP\/s double-precision performance on 64 OpenMP threads.\nAdvisor naturally provides details on the level of functions and loops, while the methods we will discuss below may require some code instrumentation in order to focus on certain code regions of interest. \n\n{\\footnotesize\n\\begin{verbatim}\n  module load advisor\/2020\n  advixe-cl --collect=roofline --project-dir=<dir>\n        -- .\/gpp 512 2 32768 20 0\n\\end{verbatim}\n}\n\n\n\\begin{figure}[h]\n\\centering\n\\makebox[0.5\\textwidth][c]{\\includegraphics[width=.5\\textwidth]{adv.png}}\n\\caption{Roofline analysis of GPP on KNL using Advisor}\n\\label{fig:adv}\n\\end{figure}\n\n\n\\subsubsection{RRZE LIKWID}\n\nLIKWID \\cite{likwid} is an open-source software package and here we use its `performance groups', FLOPS\\_DP, HBM\\_CACHE, L2 and DATA (for L1), for hierarchical Roofline data collection. \nEach of these groups contains a set of raw hardware counters and derived performance metrics, without user having to dive deep into the nitty-gritty micro-architecture specs and hardware counter details.\nThe following command can be used to profile with LIKWID, \n\n{\\footnotesize\n\\begin{verbatim}\n  module load likwid\/4.3.0\n  groups=('FLOPS_DP' 'HBM_CACHE' 'L2' 'DATA')\n  for gs in ${groups[@]}\n  do\n    likwid-perfctr -c 0-271 -g $gs \n                   .\/gpp 512 2 32768 20 0\n  done\n\\end{verbatim}\n}\n\n\\begin{figure}[h]\n\\centering\n\\makebox[0.5\\textwidth][c]{\\includegraphics[width=.5\\textwidth]{likwid.png}}\n\\caption{Roofline analysis of GPP on KNL using LIKWID}\n\\label{fig:likwid}\n\\end{figure}\n\nThe raw results for GPP are as follows, and Fig.~\\ref{fig:likwid} shows that LIKWID produces a similar Roofline chart as Advisor, with close arithmetic intensity and performance.\nThe DDR-level arithmetic intensity is extremely high in Fig.~\\ref{fig:likwid}, because the data set (1.5-2 GB) fits well into the HBM cache and there is little memory transaction between DDR and HBM.\n\n\n{\\footnotesize\n\\begin{verbatim}\n  Time: 10.2243 secs\n  GFLOPS: 5051.923 \n  MCDRAM Bytes: 742.8158 GB\n  DDR Bytes: 0.8883 GB\n  L2 Bytes: 1387.739 GB\n  L1 Bytes: 6456.799 GB\n\\end{verbatim}\n}\n\n\n\n\n\\subsubsection{Intel VTune and Intel SDE}\n\nThis is a methodology developed a few years before the full integration of Roofline into Advisor, and may still present value to users who would like to investigate the underlying details.\nIn this case, the SDE tool can be used for collection of the FLOPs count and L1 data movement, while VTune can be used for uncore data movement collection.\nThe commands and results for GPP analysis in this paper are listed below, and Fig.~\\ref{fig:sdevtune}  presents the combined data with a very high consistency with the results in Fig.~\\ref{fig:likwid} (albeit the missing L2 data). \n\n{\\footnotesize\n\\begin{verbatim}\n  # commands for SDE\n  sde64 -knl -d -iform 1 -omix result.sde \n        -global_region \n        -start_ssc_mark 111:repeat \n        -stop_ssc_mark 222:repeat \n        -- .\/gpp 512 2 32768 20 0\n\\end{verbatim}\n}\n{\\footnotesize\n\\begin{verbatim}\n  # results from SDE\n  GFLOPS: 5839.811\n  L1 Bytes: 3795.623\n\\end{verbatim}\n}\n{\\footnotesize\n\\begin{verbatim}\n  # commands for VTune\n  module load vtune\/2020\n  vtune -start-paused -r my-vtune.knl \n        -collect memory-access \n        -finalization-mode=none \n        -data-limit=0 \n        -- .\/gpp 512 2 32768 20 0\n  vtune -report hw-events \n        -group-by=package \n        -r my-vtune.knl\/ \n        -format csv -csv-delimiter comma  \n        > advisor.html\n\\end{verbatim}\n}\n{\\footnotesize\n\\begin{verbatim}\n  # results from VTune\n  DDR Bytes: 0.735\n  MCDRAM Bytes: 594.562\n\\end{verbatim}\n}\n\n\n\\begin{figure}[h]\n\\centering\n\\makebox[0.5\\textwidth][c]{\\includegraphics[width=.5\\textwidth]{sde-vtune.png}}\n\\caption{Roofline analysis of GPP on KNL using SDE and VTune}\n\\label{fig:sdevtune}\n\\end{figure}\n\n\n\\subsection{Roofline Data Collection on NVIDIA GPUs\\label{sec:metho2}}\n\nOn NVIDIA GPUs, an nvprof \\cite{nvprof} based methodology was first proposed in  \\cite{yang2019hierarchical}, then an Nsight Compute \\cite{nsight_compute} metrics based one developed in \\cite{wang2020pmbs,metho}. These methodologies require a dozen of metrics to be collected for hierarchical Roofline analysis, and could incur significant profiling overhead when the number of kernels in the code is high.\nWith nvprof phasing out in the developer toolchain, Nsight Compute has become the focus of the development of Roofline data collection methodology. \nA more simplified set of metrics are identified and validated in \\cite{metho,yang20208}, and it has since been integrated into Nsight Compute 2020 (CUDA 11 release) \\cite{ncu}.\nThe default Roofline feature shipped in Nsight Compute 2020 only includes the HBM level analysis, but it can be extended by using custom section files and\/or job scripts such as \\cite{metho,yang20208}, for hierarchical Roofline analysis.\n\n\n\\subsubsection{Custom Section Files in Nsight Compute 2020\\label{sec:ncuprof}}\nNsight Compute uses Google Protocol Buffer messages for the section file, \nand it allows users to quickly create custom section files for their own tailored analysis. \nThe following is an example in \\cite{examplescripts} that can be used to collect the hierarchical double precision Roofline data for GPP, and its results are shown in Fig.~\\ref{fig:ncuprof}.\nThe 13 FLOPs\/Byte arithmetic intensity shows that this kernel has well entered the compute bound region on the HBM level, and particular attention should be paid to the utilization of compute resources such as threads and instructions, rather than the memory system.\n\n{\\footnotesize\n\\begin{verbatim}\n  module load nsight-compute\/2020.1.0\n  ncu -k NumBandNgpown_kernel \n      -o ncu.prof \n      --section-folder .\/ncu-section-files \n      --section \n      SpeedOfLight_HierarchicalDoubleRooflineChart \n      .\/gpp 512 2 32768 20 0 \n\\end{verbatim}\n}\n\n\\begin{figure}[h]\n\\centering\n\\makebox[0.5\\textwidth][c]{\\includegraphics[width=.5\\textwidth]{ncuprof.png}}\n\\caption{Roofline analysis of GPP on V100 using Nsight Compute 2020}\n\\label{fig:ncuprof}\n\\end{figure}\n\n\n\\begin{table}[h]\n\\caption{{nvprof} metrics for Roofline data collection\n\\label{tab:nvprof}\n \n{\\footnotesize\n  \\begin{tabular}{|c|c|}\n  \\hline\n    & \\textbf{Commands\/Metrics}\\\\\n  \\hline\\hline\n\\multirow{1}{*}{Time}\n    & nvprof --print-gpu-summary .\/gpp 512 2 32768 20 0\\\\\n\\hline\\hline\n\\multirow{1}{*}{FP64 FLOPs}\n    & nvprof --metrics flop\\_count\\_dp \\\\\n    \\hline\n\\multirow{1}{*}{ FP32 FLOPs }\n    & flop\\_count\\_sp \\\\\n\\hline\n\\multirow{1}{*}{ FP16 FLOPs }\n    & flop\\_count\\_hp \\\\\n\\hline\n    Tensor Core & tensor\\_precision\\_fu\\_utilization\\\\\n\\hline\\hline\n\\multirow{3}{*}{L1 Cache}\n         & gld\\_transactions, gst\\_transactions, atomic\\_transactions\\\\ \n         & local\\_load\\_transactions, local\\_store\\_transactions \\\\\n         & shared\\_load\\_transactions, shared\\_store\\_transactions \\\\\n\\hline\n\\multirow{1}{*}{L2 Cache}\n    & l2\\_read\\_transactions, l2\\_write\\_transactions\\\\\n\\hline\n\\multirow{1}{*}{HBM}\n    & dram\\_read\\_transactions, dram\\_write\\_transactions\\\\\n\\hline\n  \\end{tabular}\n }\n\\end{table}\n\n\n\\subsubsection{The nvprof Profiler}\n\nMany developers started their GPU optimization with the nvprof profiler and our initial Roofline methodology also starts with the metrics in nvprof.\nTab.~\\ref{tab:nvprof} lists a set of metrics that can be used for hierarchical Roofline analysis and they are put in three categories, runtime, FLOPs count, and data movement (in bytes) between different memory\/cache levels.\nThese metrics are based on CUPTI and can be mapped to the PerfWorks framework in Nsight Compute through \\cite{metricsmapping}, with certain validation.\nThe following command has been used for the GPP data collection and the results are in Fig.~\\ref{fig:nvprof}, with a very similar set of arithmetic intensities on L1, L2 and HBM levels, and GFLOP\/s performance to those in Fig.~\\ref{fig:ncuprof}.\n\n{\\footnotesize\n\\begin{verbatim}\n  module load cuda\/10.2.89\n  metrics='fp_count_dp,...' # see Tab. I\n  nvprof --kernels NumBandNgpown_kernel\n         --metrics $metrics \n         .\/gpp 512 2 32768 20 0\n\\end{verbatim}\n}\n\n\n\n\n\n\\begin{figure}[h]\n\\centering\n\\makebox[0.5\\textwidth][c]{\\includegraphics[width=.5\\textwidth]{nvprof.png}}\n\\caption{Roofline analysis of GPP on V100 using nvprof metrics}\n\\label{fig:nvprof}\n\\end{figure}\n\n\n\\subsubsection{Metrics in Nsight Compute 2019}\n\nAs nvprof phases out, we have developed a data collection methodology based on Nsight Compute 2019.\nThese metrics as listed in Tab.~\\ref{tab:ncu10} are more detailed than those in nvprof, and they produce comparable results as seen in Fig.~\\ref{fig:ncu10} and Fig.~\\ref{fig:ncuprof}.\nThe commands used to collect Roofline data for GPP are as follows.\n\n{\\footnotesize\n\\begin{verbatim}\n  module load cuda\/10.2.89\n  metrics='sm__cycles_elapsed.avg,...' # see Tab. II\n  nv-nsight-cu-cli  -k NumBandNgpown_kernel\n                    --metrics $metrics \n                    .\/gpp 512 2 32768 20 0 \n\\end{verbatim}\n}\n\n\\begin{table}[h]\n\\caption{Nsight Compute 2019 metrics for Roofline data collection}\n\\label{tab:ncu10}\n \n{\\footnotesize\n  \\begin{tabular}{|@{\\;}c@{\\;\\;}|@{\\;\\;}c@{\\;}|}\n  \\hline\n    & \\textbf{Metrics}\\\\\n  \\hline\\hline\n\\multirow{2}{*}{Time}\n    & sm\\_\\_cycles\\_elapsed.avg\\\\\n    & sm\\_\\_cycles\\_elapsed.avg.per\\_second\\\\\n\\hline\\hline\n\\multirow{3}{*}{FP64 FLOPs}\n    & sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_dadd\\_pred\\_on.sum\\\\\n    & sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_dmul\\_pred\\_on.sum\\\\\n    & sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_dfma\\_pred\\_on.sum\\\\\n    \\hline\n\\multirow{3}{*}{FP32 FLOPs}\n    & sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_fadd\\_pred\\_on.sum\\\\\n    & sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_fmul\\_pred\\_on.sum\\\\\n    & sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_ffma\\_pred\\_on.sum \\\\\n\\hline\n\\multirow{3}{*}{FP16 FLOPs}\n    & sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_hadd\\_pred\\_on.sum\\\\\n    & sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_hmul\\_pred\\_on.sum\\\\\n    & sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_hfma\\_pred\\_on.sum\\\\\n\\hline\n    Tensor Core & sm\\_\\_inst\\_executed\\_pipe\\_tensor.sum\\\\\n\\hline\\hline\n\\multirow{10}{*}{L1 Cache}\n    & l1tex\\_\\_t\\_sectors\\_pipe\\_lsu\\_mem\\_global\\_op\\_ld.sum\\\\\n    & l1tex\\_\\_t\\_bytes\\_pipe\\_lsu\\_mem\\_global\\_op\\_st.sum\\\\\n    & l1tex\\_\\_t\\_set\\_accesses\\_pipe\\_lsu\\_mem\\_global\\_op\\_atom.sum\\\\\n    & l1tex\\_\\_t\\_set\\_accesses\\_pipe\\_lsu\\_mem\\_global\\_op\\_red.sum\\\\\n    & l1tex\\_\\_t\\_set\\_accesses\\_pipe\\_tex\\_mem\\_surface\\_op\\_atom.sum\\\\\n    & l1tex\\_\\_t\\_set\\_accesses\\_pipe\\_tex\\_mem\\_surface\\_op\\_red.sum\\\\\n    & l1tex\\_\\_t\\_sectors\\_pipe\\_lsu\\_mem\\_local\\_op\\_ld.sum\\\\\n    & l1tex\\_\\_t\\_sectors\\_pipe\\_lsu\\_mem\\_local\\_op\\_st.sum\\\\\n    & l1tex\\_\\_data\\_pipe\\_lsu\\_wavefronts\\_mem\\_shared\\_op\\_ld.sum\\\\\n    & l1tex\\_\\_data\\_pipe\\_lsu\\_wavefronts\\_mem\\_shared\\_op\\_st.sum\\\\\n\\hline\n\\multirow{4}{*}{L2 Cache}\n    & lts\\_\\_t\\_sectors\\_op\\_read.sum\\\\\n    & lts\\_\\_t\\_sectors\\_op\\_write.sum\\\\\n    & lts\\_\\_t\\_sectors\\_op\\_atom.sum\\\\\n    & lts\\_\\_t\\_sectors\\_op\\_red.sum\\\\\n\\hline\n\\multirow{2}{*}{HBM}\n    & dram\\_\\_sectors\\_read.sum\\\\\n    & dram\\_\\_sectors\\_write.sum\\\\\n\\hline\n  \\end{tabular}\n  }\n\\end{table}\n\n\n\n\\begin{figure}[h]\n\\centering\n\\makebox[0.5\\textwidth][c]{\\includegraphics[width=.5\\textwidth]{ncu10.png}}\n\\caption{Roofline analysis of GPP on V100 using Nsight Compute 2019 metrics}\n\\label{fig:ncu10}\n\\end{figure}\n\n\n\n\\begin{table}[h]\n\\caption{Nsight Compute 2020 metrics for Roofline data collection\n\\label{tab:ncu11}\n \n{\\footnotesize\n  \\begin{tabular}{|c|c|}\n  \\hline\n    & \\textbf{Commands\/Metrics}\\\\\n  \\hline\\hline\n\\multirow{2}{*}{Time}\n    & sm\\_\\_cycles\\_elapsed.avg \\\\ \n    & sm\\_\\_cycles\\_elapsed.avg.per\\_second \\\\\n\\hline\\hline\n\\multirow{3}{*}{FP64 FLOPs}\n& sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_dadd\\_pred\\_on.sum \\\\\n& sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_dfma\\_pred\\_on.sum \\\\\n& sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_dmul\\_pred\\_on.sum \\\\\n    \\hline\n\\multirow{3}{*}{ FP32 FLOPs }\n    & sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_fadd\\_pred\\_on.sum \\\\\n    & sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_ffma\\_pred\\_on.sum \\\\\n    & sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_fmul\\_pred\\_on.sum \\\\\n\\hline\n\\multirow{3}{*}{ FP16 FLOPs }\n    & sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_hadd\\_pred\\_on.sum \\\\\n    & sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_hfma\\_pred\\_on.sum \\\\\n    & sm\\_\\_sass\\_thread\\_inst\\_executed\\_op\\_hmul\\_pred\\_on.sum \\\\\n\\hline\n     Tensor Core  & \\multirow{1}{*}{sm\\_\\_inst\\_executed\\_pipe\\_tensor.sum}\\\\\n   \n\\hline\\hline\n\\multirow{1}{*}{L1 Cache}\n         & l1tex\\_\\_t\\_bytes.sum \\\\\n\\hline\n\\multirow{1}{*}{L2 Cache}\n    & lts\\_\\_t\\_bytes.sum \\\\\n\\hline\n\\multirow{1}{*}{HBM}\n    & dram\\_\\_bytes.sum\\\\\n\\hline\n  \\end{tabular}\n }\n\\end{table}\n\n\n\\subsubsection{Metrics in Nsight Compute 2020}\n\nAs Nsight Compute evolves over time, we have also developed a more simplified data collection methodology with fewer metrics to collect (please see Tab.~\\ref{tab:ncu11}).\nThese metrics are equivalent to the ones used in section files in \\ref{sec:ncuprof}, and scripts based on them \\cite{metho} can be used for easier integration with users' other job submission workflows, and for more customized Roofline presentation (using Matplotlib).\nThe commands we used to collect Roofline information for GPP in this paper are as follows.\n\n \n{\\footnotesize\n\\begin{verbatim}\n  module load nsight-compute\/2020.1.0\n  metrics='sm__cycles_elapsed.avg,...'\n  ncu -k NumBandNgpown_kernel\n      --metrics $metrics \n      .\/gpp 512 2 32768 20 0  \n\\end{verbatim}\n}\n\nFig.~\\ref{fig:ncu11} shows that this methodology produces consistent results as in previous subsections, with very \nmarginal difference on the arithmetic intensity and GFLOP\/s throughput.\n\n\n\n\n\n\n \n\n\\begin{figure}[h]\n\\centering\n\\makebox[0.5\\textwidth][c]{\\includegraphics[width=.5\\textwidth]{ncu11.png}}\n\\caption{Roofline analysis of GPP on V100 using Nsight Compute 2020 metrics}\n\\label{fig:ncu11}\n\\end{figure}\n\n\n\n\n\\section{Summary}\n\nIn this paper, we have presented a range of methods using a variety of performance tools to collect hierarchical data for Roofline analysis. \nEven though the Roofline model has been integrated into production tools such as Intel Advisor and NVIDIA Nsight Compute, we still expect that this paper fills the gaps for developers who do not have access to those tools, or who would like to investigate the underlying details.\nIt would serve the purpose of flexibility and generality in the Roofline data collection space.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}